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<div class="titlepage"><div><div><h3 class="title">
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<a name="math_toolkit.sf_beta.beta_function"></a><a class="link" href="beta_function.html" title="Beta">Beta</a>
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</h3></div></div></div>
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<h5>
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<a name="math_toolkit.sf_beta.beta_function.h0"></a>
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<span class="phrase"><a name="math_toolkit.sf_beta.beta_function.synopsis"></a></span><a class="link" href="beta_function.html#math_toolkit.sf_beta.beta_function.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">beta</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">T1</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">T2</span><span class="special">></span>
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<a class="link" href="../result_type.html" title="Calculation of the Type of the Result"><span class="emphasis"><em>calculated-result-type</em></span></a> <span class="identifier">beta</span><span class="special">(</span><span class="identifier">T1</span> <span class="identifier">a</span><span class="special">,</span> <span class="identifier">T2</span> <span class="identifier">b</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">T1</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">T2</span><span class="special">,</span> <span class="keyword">class</span> <a class="link" href="../../policy.html" title="Chapter 22. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">></span>
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<a class="link" href="../result_type.html" title="Calculation of the Type of the Result"><span class="emphasis"><em>calculated-result-type</em></span></a> <span class="identifier">beta</span><span class="special">(</span><span class="identifier">T1</span> <span class="identifier">a</span><span class="special">,</span> <span class="identifier">T2</span> <span class="identifier">b</span><span class="special">,</span> <span class="keyword">const</span> <a class="link" href="../../policy.html" title="Chapter 22. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&);</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.sf_beta.beta_function.h1"></a>
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<span class="phrase"><a name="math_toolkit.sf_beta.beta_function.description"></a></span><a class="link" href="beta_function.html#math_toolkit.sf_beta.beta_function.description">Description</a>
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</h5>
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<p>
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The beta function is defined by:
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</p>
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<div class="blockquote"><blockquote class="blockquote"><p>
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<span class="inlinemediaobject"><img src="../../../equations/beta1.svg"></span>
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</p></blockquote></div>
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<div class="blockquote"><blockquote class="blockquote"><p>
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<span class="inlinemediaobject"><img src="../../../graphs/beta.svg" align="middle"></span>
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</p></blockquote></div>
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<p>
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The final <a class="link" href="../../policy.html" title="Chapter 22. 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. Refer to the <a class="link" href="../../policy.html" title="Chapter 22. Policies: Controlling Precision, Error Handling etc">policy
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documentation for more details</a>.
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</p>
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<p>
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The return type of these functions is computed using the <a class="link" href="../result_type.html" title="Calculation of the Type of the Result"><span class="emphasis"><em>result
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type calculation rules</em></span></a> when T1 and T2 are different types.
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</p>
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<h5>
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<a name="math_toolkit.sf_beta.beta_function.h2"></a>
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<span class="phrase"><a name="math_toolkit.sf_beta.beta_function.accuracy"></a></span><a class="link" href="beta_function.html#math_toolkit.sf_beta.beta_function.accuracy">Accuracy</a>
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</h5>
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<p>
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The following table shows peak errors for various domains of input arguments,
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along with comparisons to the <a href="http://www.gnu.org/software/gsl/" target="_top">GSL-1.9</a>
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and <a href="http://www.netlib.org/cephes/" target="_top">Cephes</a> libraries.
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Note that only results for the widest floating point type on the system are
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given as narrower types have <a class="link" href="../relative_error.html#math_toolkit.relative_error.zero_error">effectively
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zero error</a>.
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</p>
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<div class="table">
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<a name="math_toolkit.sf_beta.beta_function.table_beta"></a><p class="title"><b>Table 8.17. Error rates for beta</b></p>
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<div class="table-contents"><table class="table" summary="Error rates for beta">
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<colgroup>
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<col>
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<col>
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<col>
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<col>
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<col>
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</colgroup>
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<thead><tr>
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<th>
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</th>
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<th>
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<p>
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GNU C++ version 7.1.0<br> linux<br> double
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</p>
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</th>
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<th>
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<p>
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GNU C++ version 7.1.0<br> linux<br> long double
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</p>
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</th>
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<th>
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<p>
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Sun compiler version 0x5150<br> Sun Solaris<br> long double
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</p>
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</th>
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<th>
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<p>
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Microsoft Visual C++ version 14.1<br> Win32<br> double
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</p>
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</th>
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</tr></thead>
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<tbody>
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<tr>
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<td>
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<p>
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Beta Function: Small Values
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</p>
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</td>
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<td>
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<p>
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<span class="blue">Max = 0ε (Mean = 0ε)</span><br> <br> (<span class="emphasis"><em>GSL
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2.1:</em></span> <span class="red">Max = +INFε (Mean = +INFε) <a class="link" href="../logs_and_tables/logs.html#errors_GNU_C_version_7_1_0_linux_double_beta_GSL_2_1_Beta_Function_Small_Values">And
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other failures.</a>)</span><br> (<span class="emphasis"><em>Rmath 3.2.3:</em></span>
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Max = 1.14ε (Mean = 0.574ε))
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</p>
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</td>
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<td>
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<p>
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<span class="blue">Max = 2.86ε (Mean = 1.22ε)</span><br> <br>
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(<span class="emphasis"><em><cmath>:</em></span> Max = 364ε (Mean = 76.2ε))
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</p>
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</td>
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<td>
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<p>
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<span class="blue">Max = 2.86ε (Mean = 1.22ε)</span>
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</p>
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</td>
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<td>
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<p>
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<span class="blue">Max = 2.23ε (Mean = 1.14ε)</span>
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</p>
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</td>
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</tr>
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<tr>
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<td>
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<p>
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Beta Function: Medium Values
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</p>
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</td>
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<td>
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<p>
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<span class="blue">Max = 0.978ε (Mean = 0.0595ε)</span><br>
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<br> (<span class="emphasis"><em>GSL 2.1:</em></span> Max = 1.18e+03ε (Mean = 238ε))<br>
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(<span class="emphasis"><em>Rmath 3.2.3:</em></span> Max = 1.09e+03ε (Mean = 265ε))
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</p>
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</td>
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<td>
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<p>
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<span class="blue">Max = 61.4ε (Mean = 19.4ε)</span><br> <br>
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(<span class="emphasis"><em><cmath>:</em></span> Max = 1.07e+03ε (Mean = 264ε))
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</p>
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</td>
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<td>
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<p>
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<span class="blue">Max = 107ε (Mean = 24.5ε)</span>
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</p>
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</td>
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<td>
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<p>
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<span class="blue">Max = 96.5ε (Mean = 22.4ε)</span>
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</p>
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</td>
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</tr>
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<tr>
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<td>
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<p>
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Beta Function: Divergent Values
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</p>
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</td>
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<td>
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<p>
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<span class="blue">Max = 0ε (Mean = 0ε)</span><br> <br> (<span class="emphasis"><em>GSL
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2.1:</em></span> Max = 12.1ε (Mean = 1.99ε))<br> (<span class="emphasis"><em>Rmath
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3.2.3:</em></span> Max = 176ε (Mean = 28ε))
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</p>
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</td>
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<td>
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<p>
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<span class="blue">Max = 8.99ε (Mean = 2.44ε)</span><br> <br>
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(<span class="emphasis"><em><cmath>:</em></span> Max = 128ε (Mean = 23.8ε))
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</p>
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</td>
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<td>
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<p>
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<span class="blue">Max = 18.8ε (Mean = 2.71ε)</span>
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</p>
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</td>
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<td>
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<p>
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<span class="blue">Max = 11.4ε (Mean = 2.19ε)</span>
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</p>
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</td>
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</tr>
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</tbody>
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</table></div>
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</div>
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<br class="table-break"><p>
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Note that the worst errors occur when a or b are large, and that when this
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is the case the result is very close to zero, so absolute errors will be
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very small.
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</p>
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<h5>
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<a name="math_toolkit.sf_beta.beta_function.h3"></a>
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<span class="phrase"><a name="math_toolkit.sf_beta.beta_function.testing"></a></span><a class="link" href="beta_function.html#math_toolkit.sf_beta.beta_function.testing">Testing</a>
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</h5>
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<p>
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A mixture of spot tests of exact values, and randomly generated test data
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are used: the test data was computed using <a href="http://shoup.net/ntl/doc/RR.txt" target="_top">NTL::RR</a>
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at 1000-bit precision.
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</p>
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<h5>
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<a name="math_toolkit.sf_beta.beta_function.h4"></a>
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<span class="phrase"><a name="math_toolkit.sf_beta.beta_function.implementation"></a></span><a class="link" href="beta_function.html#math_toolkit.sf_beta.beta_function.implementation">Implementation</a>
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</h5>
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<p>
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Traditional methods of evaluating the beta function either involve evaluating
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the gamma functions directly, or taking logarithms and then exponentiating
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the result. However, the former is prone to overflows for even very modest
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arguments, while the latter is prone to cancellation errors. As an alternative,
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if we regard the gamma function as a white-box containing the <a class="link" href="../lanczos.html" title="The Lanczos Approximation">Lanczos
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approximation</a>, then we can combine the power terms:
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</p>
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<div class="blockquote"><blockquote class="blockquote"><p>
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<span class="inlinemediaobject"><img src="../../../equations/beta2.svg"></span>
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</p></blockquote></div>
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<p>
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which is almost the ideal solution, however almost all of the error occurs
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in evaluating the power terms when <span class="emphasis"><em>a</em></span> or <span class="emphasis"><em>b</em></span>
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are large. If we assume that <span class="emphasis"><em>a > b</em></span> then the larger
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of the two power terms can be reduced by a factor of <span class="emphasis"><em>b</em></span>,
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which immediately cuts the maximum error in half:
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</p>
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<div class="blockquote"><blockquote class="blockquote"><p>
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<span class="inlinemediaobject"><img src="../../../equations/beta3.svg"></span>
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</p></blockquote></div>
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<p>
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This may not be the final solution, but it is very competitive compared to
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other implementation methods.
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</p>
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<p>
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The generic implementation - where no <a class="link" href="../lanczos.html" title="The Lanczos Approximation">Lanczos
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approximation</a> approximation is available - is implemented in a very
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similar way to the generic version of the gamma function by means of Sterling's
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approximation. Again in order to avoid numerical overflow the power terms
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that prefix the series are collected together
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</p>
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<p>
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There are a few special cases worth mentioning:
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</p>
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<p>
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When <span class="emphasis"><em>a</em></span> or <span class="emphasis"><em>b</em></span> are less than one,
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we can use the recurrence relations:
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</p>
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<div class="blockquote"><blockquote class="blockquote"><p>
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<span class="inlinemediaobject"><img src="../../../equations/beta4.svg"></span>
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</p></blockquote></div>
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<div class="blockquote"><blockquote class="blockquote"><p>
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<span class="inlinemediaobject"><img src="../../../equations/beta5.svg"></span>
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</p></blockquote></div>
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<p>
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to move to a more favorable region where they are both greater than 1.
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</p>
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<p>
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In addition:
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</p>
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<div class="blockquote"><blockquote class="blockquote"><p>
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<span class="inlinemediaobject"><img src="../../../equations/beta7.svg"></span>
|
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|
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</p></blockquote></div>
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</div>
|
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<div class="copyright-footer">Copyright © 2006-2021 Nikhar Agrawal, Anton Bikineev, Matthew Borland,
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Paul A. Bristow, Marco Guazzone, Christopher Kormanyos, Hubert Holin, Bruno
|
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Lalande, John Maddock, Evan Miller, 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>)
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</p>
|
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</div>
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<hr>
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