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  25 <div class="section">
  26 <div class="titlepage"><div><div><h3 class="title">
  27 <a name="math_toolkit.sf_beta.ibeta_function"></a><a class="link" href="ibeta_function.html" title="Incomplete Beta Functions">Incomplete Beta
  28       Functions</a>
  29 </h3></div></div></div>
  30 <h5>
  31 <a name="math_toolkit.sf_beta.ibeta_function.h0"></a>
  32         <span class="phrase"><a name="math_toolkit.sf_beta.ibeta_function.synopsis"></a></span><a class="link" href="ibeta_function.html#math_toolkit.sf_beta.ibeta_function.synopsis">Synopsis</a>
  33       </h5>
  34 <pre class="programlisting"><span class="preprocessor">#include</span> <span class="special">&lt;</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">&gt;</span>
  35 </pre>
  36 <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>
  37
  38 <span class="keyword">template</span> <span class="special">&lt;</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> <span class="identifier">T3</span><span class="special">&gt;</span>
  39 <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">ibeta</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="identifier">T3</span> <span class="identifier">x</span><span class="special">);</span>
  40
  41 <span class="keyword">template</span> <span class="special">&lt;</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> <span class="identifier">T3</span><span class="special">,</span> <span class="keyword">class</span> <a class="link" href="../../policy.html" title="Chapter&#160;20.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&gt;</span>
  42 <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">ibeta</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="identifier">T3</span> <span class="identifier">x</span><span class="special">,</span> <span class="keyword">const</span> <a class="link" href="../../policy.html" title="Chapter&#160;20.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&amp;);</span>
  43
  44 <span class="keyword">template</span> <span class="special">&lt;</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> <span class="identifier">T3</span><span class="special">&gt;</span>
  45 <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">ibetac</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="identifier">T3</span> <span class="identifier">x</span><span class="special">);</span>
  46
  47 <span class="keyword">template</span> <span class="special">&lt;</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> <span class="identifier">T3</span><span class="special">,</span> <span class="keyword">class</span> <a class="link" href="../../policy.html" title="Chapter&#160;20.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&gt;</span>
  48 <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">ibetac</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="identifier">T3</span> <span class="identifier">x</span><span class="special">,</span> <span class="keyword">const</span> <a class="link" href="../../policy.html" title="Chapter&#160;20.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&amp;);</span>
  49
  50 <span class="keyword">template</span> <span class="special">&lt;</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> <span class="identifier">T3</span><span class="special">&gt;</span>
  51 <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="identifier">T3</span> <span class="identifier">x</span><span class="special">);</span>
  52
  53 <span class="keyword">template</span> <span class="special">&lt;</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> <span class="identifier">T3</span><span class="special">,</span> <span class="keyword">class</span> <a class="link" href="../../policy.html" title="Chapter&#160;20.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&gt;</span>
  54 <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="identifier">T3</span> <span class="identifier">x</span><span class="special">,</span> <span class="keyword">const</span> <a class="link" href="../../policy.html" title="Chapter&#160;20.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&amp;);</span>
  55
  56 <span class="keyword">template</span> <span class="special">&lt;</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> <span class="identifier">T3</span><span class="special">&gt;</span>
  57 <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">betac</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="identifier">T3</span> <span class="identifier">x</span><span class="special">);</span>
  58
  59 <span class="keyword">template</span> <span class="special">&lt;</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> <span class="identifier">T3</span><span class="special">,</span> <span class="keyword">class</span> <a class="link" href="../../policy.html" title="Chapter&#160;20.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&gt;</span>
  60 <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">betac</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="identifier">T3</span> <span class="identifier">x</span><span class="special">,</span> <span class="keyword">const</span> <a class="link" href="../../policy.html" title="Chapter&#160;20.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&amp;);</span>
  61
  62 <span class="special">}}</span> <span class="comment">// namespaces</span>
  63 </pre>
  64 <h5>
  65 <a name="math_toolkit.sf_beta.ibeta_function.h1"></a>
  66         <span class="phrase"><a name="math_toolkit.sf_beta.ibeta_function.description"></a></span><a class="link" href="ibeta_function.html#math_toolkit.sf_beta.ibeta_function.description">Description</a>
  67       </h5>
  68 <p>
  69         There are four <a href="http://en.wikipedia.org/wiki/Incomplete_beta_function" target="_top">incomplete
  70         beta functions</a> : two are normalised versions (also known as <span class="emphasis"><em>regularized</em></span>
  71         beta functions) that return values in the range [0, 1], and two are non-normalised
  72         and return values in the range [0, <a class="link" href="beta_function.html" title="Beta">beta</a>(a,
  73         b)]. Users interested in statistical applications should use the normalised
  74         (or <a href="http://mathworld.wolfram.com/RegularizedBetaFunction.html" target="_top">regularized</a>
  75         ) versions (ibeta and ibetac).
  76       </p>
  77 <p>
  78         All of these functions require <span class="emphasis"><em>0 &lt;= x &lt;= 1</em></span>.
  79       </p>
  80 <p>
  81         The normalized functions <a class="link" href="ibeta_function.html" title="Incomplete Beta Functions">ibeta</a>
  82         and <a class="link" href="ibeta_function.html" title="Incomplete Beta Functions">ibetac</a> require
  83         <span class="emphasis"><em>a,b &gt;= 0</em></span>, and in addition that not both <span class="emphasis"><em>a</em></span>
  84         and <span class="emphasis"><em>b</em></span> are zero.
  85       </p>
  86 <p>
  87         The functions <a class="link" href="beta_function.html" title="Beta">beta</a>
  88         and <a class="link" href="ibeta_function.html" title="Incomplete Beta Functions">betac</a> require
  89         <span class="emphasis"><em>a,b &gt; 0</em></span>.
  90       </p>
  91 <p>
  92         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
  93         type calculation rules</em></span></a> when T1, T2 and T3 are different
  94         types.
  95       </p>
  96 <p>
  97         The final <a class="link" href="../../policy.html" title="Chapter&#160;20.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a> argument is optional and can
  98         be used to control the behaviour of the function: how it handles errors,
  99         what level of precision to use etc. Refer to the <a class="link" href="../../policy.html" title="Chapter&#160;20.&#160;Policies: Controlling Precision, Error Handling etc">policy
 100         documentation for more details</a>.
 101       </p>
 102 <pre class="programlisting"><span class="keyword">template</span> <span class="special">&lt;</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> <span class="identifier">T3</span><span class="special">&gt;</span>
 103 <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">ibeta</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="identifier">T3</span> <span class="identifier">x</span><span class="special">);</span>
 104
 105 <span class="keyword">template</span> <span class="special">&lt;</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> <span class="identifier">T3</span><span class="special">,</span> <span class="keyword">class</span> <a class="link" href="../../policy.html" title="Chapter&#160;20.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&gt;</span>
 106 <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">ibeta</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="identifier">T3</span> <span class="identifier">x</span><span class="special">,</span> <span class="keyword">const</span> <a class="link" href="../../policy.html" title="Chapter&#160;20.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&amp;);</span>
 107 </pre>
 108 <p>
 109         Returns the normalised incomplete beta function of a, b and x:
 110       </p>
 111 <div class="blockquote"><blockquote class="blockquote"><p>
 112           <span class="inlinemediaobject"><img src="../../../equations/ibeta3.svg"></span>
 113
 114         </p></blockquote></div>
 115 <div class="blockquote"><blockquote class="blockquote"><p>
 116           <span class="inlinemediaobject"><img src="../../../graphs/ibeta.svg" align="middle"></span>
 117
 118         </p></blockquote></div>
 119 <pre class="programlisting"><span class="keyword">template</span> <span class="special">&lt;</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> <span class="identifier">T3</span><span class="special">&gt;</span>
 120 <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">ibetac</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="identifier">T3</span> <span class="identifier">x</span><span class="special">);</span>
 121
 122 <span class="keyword">template</span> <span class="special">&lt;</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> <span class="identifier">T3</span><span class="special">,</span> <span class="keyword">class</span> <a class="link" href="../../policy.html" title="Chapter&#160;20.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&gt;</span>
 123 <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">ibetac</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="identifier">T3</span> <span class="identifier">x</span><span class="special">,</span> <span class="keyword">const</span> <a class="link" href="../../policy.html" title="Chapter&#160;20.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&amp;);</span>
 124 </pre>
 125 <p>
 126         Returns the normalised complement of the incomplete beta function of a, b
 127         and x:
 128       </p>
 129 <div class="blockquote"><blockquote class="blockquote"><p>
 130           <span class="inlinemediaobject"><img src="../../../equations/ibeta4.svg"></span>
 131
 132         </p></blockquote></div>
 133 <pre class="programlisting"><span class="keyword">template</span> <span class="special">&lt;</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> <span class="identifier">T3</span><span class="special">&gt;</span>
 134 <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="identifier">T3</span> <span class="identifier">x</span><span class="special">);</span>
 135
 136 <span class="keyword">template</span> <span class="special">&lt;</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> <span class="identifier">T3</span><span class="special">,</span> <span class="keyword">class</span> <a class="link" href="../../policy.html" title="Chapter&#160;20.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&gt;</span>
 137 <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="identifier">T3</span> <span class="identifier">x</span><span class="special">,</span> <span class="keyword">const</span> <a class="link" href="../../policy.html" title="Chapter&#160;20.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&amp;);</span>
 138 </pre>
 139 <p>
 140         Returns the full (non-normalised) incomplete beta function of a, b and x:
 141       </p>
 142 <div class="blockquote"><blockquote class="blockquote"><p>
 143           <span class="inlinemediaobject"><img src="../../../equations/ibeta1.svg"></span>
 144
 145         </p></blockquote></div>
 146 <pre class="programlisting"><span class="keyword">template</span> <span class="special">&lt;</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> <span class="identifier">T3</span><span class="special">&gt;</span>
 147 <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">betac</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="identifier">T3</span> <span class="identifier">x</span><span class="special">);</span>
 148
 149 <span class="keyword">template</span> <span class="special">&lt;</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> <span class="identifier">T3</span><span class="special">,</span> <span class="keyword">class</span> <a class="link" href="../../policy.html" title="Chapter&#160;20.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&gt;</span>
 150 <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">betac</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="identifier">T3</span> <span class="identifier">x</span><span class="special">,</span> <span class="keyword">const</span> <a class="link" href="../../policy.html" title="Chapter&#160;20.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&amp;);</span>
 151 </pre>
 152 <p>
 153         Returns the full (non-normalised) complement of the incomplete beta function
 154         of a, b and x:
 155       </p>
 156 <div class="blockquote"><blockquote class="blockquote"><p>
 157           <span class="inlinemediaobject"><img src="../../../equations/ibeta2.svg"></span>
 158
 159         </p></blockquote></div>
 160 <h5>
 161 <a name="math_toolkit.sf_beta.ibeta_function.h2"></a>
 162         <span class="phrase"><a name="math_toolkit.sf_beta.ibeta_function.accuracy"></a></span><a class="link" href="ibeta_function.html#math_toolkit.sf_beta.ibeta_function.accuracy">Accuracy</a>
 163       </h5>
 164 <p>
 165         The following tables give peak and mean relative errors in over various domains
 166         of a, b and x, along with comparisons to the <a href="http://www.gnu.org/software/gsl/" target="_top">GSL-1.9</a>
 167         and <a href="http://www.netlib.org/cephes/" target="_top">Cephes</a> libraries.
 168         Note that only results for the widest floating-point type on the system are
 169         given as narrower types have <a class="link" href="../relative_error.html#math_toolkit.relative_error.zero_error">effectively
 170         zero error</a>.
 171       </p>
 172 <p>
 173         Note that the results for 80 and 128-bit long doubles are noticeably higher
 174         than for doubles: this is because the wider exponent range of these types
 175         allow more extreme test cases to be tested. For example expected results
 176         that are zero at double precision, may be finite but exceptionally small
 177         with the wider exponent range of the long double types.
 178       </p>
 179 <div class="table">
 180 <a name="math_toolkit.sf_beta.ibeta_function.table_ibeta"></a><p class="title"><b>Table&#160;8.18.&#160;Error rates for ibeta</b></p>
 181 <div class="table-contents"><table class="table" summary="Error rates for ibeta">
 182 <colgroup>
 183 <col>
 184 <col>
 185 <col>
 186 <col>
 187 <col>
 188 </colgroup>
 189 <thead><tr>
 190 <th>
 191               </th>
 192 <th>
 193                 <p>
 194                   GNU C++ version 7.1.0<br> linux<br> double
 195                 </p>
 196               </th>
 197 <th>
 198                 <p>
 199                   GNU C++ version 7.1.0<br> linux<br> long double
 200                 </p>
 201               </th>
 202 <th>
 203                 <p>
 204                   Sun compiler version 0x5150<br> Sun Solaris<br> long double
 205                 </p>
 206               </th>
 207 <th>
 208                 <p>
 209                   Microsoft Visual C++ version 14.1<br> Win32<br> double
 210                 </p>
 211               </th>
 212 </tr></thead>
 213 <tbody>
 214 <tr>
 215 <td>
 216                 <p>
 217                   Incomplete Beta Function: Small Values
 218                 </p>
 219               </td>
 220 <td>
 221                 <p>
 222                   <span class="blue">Max = 0&#949; (Mean = 0&#949;)</span><br> <br> (<span class="emphasis"><em>GSL
 223                   2.1:</em></span> Max = 682&#949; (Mean = 32.6&#949;))<br> (<span class="emphasis"><em>Rmath
 224                   3.2.3:</em></span> Max = 22.9&#949; (Mean = 3.35&#949;))
 225                 </p>
 226               </td>
 227 <td>
 228                 <p>
 229                   <span class="blue">Max = 8.97&#949; (Mean = 2.09&#949;)</span>
 230                 </p>
 231               </td>
 232 <td>
 233                 <p>
 234                   <span class="blue">Max = 21.3&#949; (Mean = 2.75&#949;)</span>
 235                 </p>
 236               </td>
 237 <td>
 238                 <p>
 239                   <span class="blue">Max = 8.4&#949; (Mean = 1.93&#949;)</span>
 240                 </p>
 241               </td>
 242 </tr>
 243 <tr>
 244 <td>
 245                 <p>
 246                   Incomplete Beta Function: Medium Values
 247                 </p>
 248               </td>
 249 <td>
 250                 <p>
 251                   <span class="blue">Max = 0&#949; (Mean = 0&#949;)</span><br> <br> (<span class="emphasis"><em>GSL
 252                   2.1:</em></span> Max = 690&#949; (Mean = 151&#949;))<br> (<span class="emphasis"><em>Rmath 3.2.3:</em></span>
 253                   Max = 232&#949; (Mean = 27.9&#949;))
 254                 </p>
 255               </td>
 256 <td>
 257                 <p>
 258                   <span class="blue">Max = 50&#949; (Mean = 12.1&#949;)</span>
 259                 </p>
 260               </td>
 261 <td>
 262                 <p>
 263                   <span class="blue">Max = 124&#949; (Mean = 18.4&#949;)</span>
 264                 </p>
 265               </td>
 266 <td>
 267                 <p>
 268                   <span class="blue">Max = 106&#949; (Mean = 16.3&#949;)</span>
 269                 </p>
 270               </td>
 271 </tr>
 272 <tr>
 273 <td>
 274                 <p>
 275                   Incomplete Beta Function: Large and Diverse Values
 276                 </p>
 277               </td>
 278 <td>
 279                 <p>
 280                   <span class="blue">Max = 1.26&#949; (Mean = 0.063&#949;)</span><br> <br>
 281                   (<span class="emphasis"><em>GSL 2.1:</em></span> Max = 3.9e+05&#949; (Mean = 1.82e+04&#949;)
 282                   <a class="link" href="../logs_and_tables/logs.html#errors_GNU_C_version_7_1_0_linux_double_ibeta_GSL_2_1_Incomplete_Beta_Function_Large_and_Diverse_Values">And
 283                   other failures.</a>)<br> (<span class="emphasis"><em>Rmath 3.2.3:</em></span>
 284                   Max = 574&#949; (Mean = 49.4&#949;))
 285                 </p>
 286               </td>
 287 <td>
 288                 <p>
 289                   <span class="blue">Max = 1.96e+04&#949; (Mean = 997&#949;)</span>
 290                 </p>
 291               </td>
 292 <td>
 293                 <p>
 294                   <span class="blue">Max = 4.98e+04&#949; (Mean = 2.07e+03&#949;)</span>
 295                 </p>
 296               </td>
 297 <td>
 298                 <p>
 299                   <span class="blue">Max = 1.32e+03&#949; (Mean = 68.5&#949;)</span>
 300                 </p>
 301               </td>
 302 </tr>
 303 <tr>
 304 <td>
 305                 <p>
 306                   Incomplete Beta Function: Small Integer Values
 307                 </p>
 308               </td>
 309 <td>
 310                 <p>
 311                   <span class="blue">Max = 0&#949; (Mean = 0&#949;)</span><br> <br> (<span class="emphasis"><em>GSL
 312                   2.1:</em></span> Max = 254&#949; (Mean = 50.9&#949;))<br> (<span class="emphasis"><em>Rmath
 313                   3.2.3:</em></span> Max = 62.2&#949; (Mean = 8.95&#949;))
 314                 </p>
 315               </td>
 316 <td>
 317                 <p>
 318                   <span class="blue">Max = 4.45&#949; (Mean = 0.814&#949;)</span>
 319                 </p>
 320               </td>
 321 <td>
 322                 <p>
 323                   <span class="blue">Max = 44.5&#949; (Mean = 10.1&#949;)</span>
 324                 </p>
 325               </td>
 326 <td>
 327                 <p>
 328                   <span class="blue">Max = 3.85&#949; (Mean = 0.791&#949;)</span>
 329                 </p>
 330               </td>
 331 </tr>
 332 </tbody>
 333 </table></div>
 334 </div>
 335 <br class="table-break"><div class="table">
 336 <a name="math_toolkit.sf_beta.ibeta_function.table_ibetac"></a><p class="title"><b>Table&#160;8.19.&#160;Error rates for ibetac</b></p>
 337 <div class="table-contents"><table class="table" summary="Error rates for ibetac">
 338 <colgroup>
 339 <col>
 340 <col>
 341 <col>
 342 <col>
 343 <col>
 344 </colgroup>
 345 <thead><tr>
 346 <th>
 347               </th>
 348 <th>
 349                 <p>
 350                   GNU C++ version 7.1.0<br> linux<br> double
 351                 </p>
 352               </th>
 353 <th>
 354                 <p>
 355                   GNU C++ version 7.1.0<br> linux<br> long double
 356                 </p>
 357               </th>
 358 <th>
 359                 <p>
 360                   Sun compiler version 0x5150<br> Sun Solaris<br> long double
 361                 </p>
 362               </th>
 363 <th>
 364                 <p>
 365                   Microsoft Visual C++ version 14.1<br> Win32<br> double
 366                 </p>
 367               </th>
 368 </tr></thead>
 369 <tbody>
 370 <tr>
 371 <td>
 372                 <p>
 373                   Incomplete Beta Function: Small Values
 374                 </p>
 375               </td>
 376 <td>
 377                 <p>
 378                   <span class="blue">Max = 0&#949; (Mean = 0&#949;)</span><br> <br> (<span class="emphasis"><em>Rmath
 379                   3.2.3:</em></span> Max = 22.4&#949; (Mean = 3.67&#949;))
 380                 </p>
 381               </td>
 382 <td>
 383                 <p>
 384                   <span class="blue">Max = 10.6&#949; (Mean = 2.22&#949;)</span>
 385                 </p>
 386               </td>
 387 <td>
 388                 <p>
 389                   <span class="blue">Max = 13.8&#949; (Mean = 2.68&#949;)</span>
 390                 </p>
 391               </td>
 392 <td>
 393                 <p>
 394                   <span class="blue">Max = 6.94&#949; (Mean = 1.71&#949;)</span>
 395                 </p>
 396               </td>
 397 </tr>
 398 <tr>
 399 <td>
 400                 <p>
 401                   Incomplete Beta Function: Medium Values
 402                 </p>
 403               </td>
 404 <td>
 405                 <p>
 406                   <span class="blue">Max = 0&#949; (Mean = 0&#949;)</span><br> <br> (<span class="emphasis"><em>Rmath
 407                   3.2.3:</em></span> Max = 204&#949; (Mean = 25.8&#949;))
 408                 </p>
 409               </td>
 410 <td>
 411                 <p>
 412                   <span class="blue">Max = 73.9&#949; (Mean = 11.2&#949;)</span>
 413                 </p>
 414               </td>
 415 <td>
 416                 <p>
 417                   <span class="blue">Max = 132&#949; (Mean = 19.8&#949;)</span>
 418                 </p>
 419               </td>
 420 <td>
 421                 <p>
 422                   <span class="blue">Max = 56.7&#949; (Mean = 14.3&#949;)</span>
 423                 </p>
 424               </td>
 425 </tr>
 426 <tr>
 427 <td>
 428                 <p>
 429                   Incomplete Beta Function: Large and Diverse Values
 430                 </p>
 431               </td>
 432 <td>
 433                 <p>
 434                   <span class="blue">Max = 0.981&#949; (Mean = 0.0573&#949;)</span><br>
 435                   <br> (<span class="emphasis"><em>Rmath 3.2.3:</em></span> Max = 889&#949; (Mean = 68.4&#949;))
 436                 </p>
 437               </td>
 438 <td>
 439                 <p>
 440                   <span class="blue">Max = 3.45e+04&#949; (Mean = 1.32e+03&#949;)</span>
 441                 </p>
 442               </td>
 443 <td>
 444                 <p>
 445                   <span class="blue">Max = 6.31e+04&#949; (Mean = 2.04e+03&#949;)</span>
 446                 </p>
 447               </td>
 448 <td>
 449                 <p>
 450                   <span class="blue">Max = 1.88e+03&#949; (Mean = 82.7&#949;)</span>
 451                 </p>
 452               </td>
 453 </tr>
 454 <tr>
 455 <td>
 456                 <p>
 457                   Incomplete Beta Function: Small Integer Values
 458                 </p>
 459               </td>
 460 <td>
 461                 <p>
 462                   <span class="blue">Max = 0&#949; (Mean = 0&#949;)</span><br> <br> (<span class="emphasis"><em>Rmath
 463                   3.2.3:</em></span> Max = 84.6&#949; (Mean = 18&#949;))
 464                 </p>
 465               </td>
 466 <td>
 467                 <p>
 468                   <span class="blue">Max = 5.34&#949; (Mean = 1.11&#949;)</span>
 469                 </p>
 470               </td>
 471 <td>
 472                 <p>
 473                   <span class="blue">Max = 107&#949; (Mean = 17.1&#949;)</span>
 474                 </p>
 475               </td>
 476 <td>
 477                 <p>
 478                   <span class="blue">Max = 6.37&#949; (Mean = 1.03&#949;)</span>
 479                 </p>
 480               </td>
 481 </tr>
 482 </tbody>
 483 </table></div>
 484 </div>
 485 <br class="table-break"><div class="table">
 486 <a name="math_toolkit.sf_beta.ibeta_function.table_beta_incomplete_"></a><p class="title"><b>Table&#160;8.20.&#160;Error rates for beta (incomplete)</b></p>
 487 <div class="table-contents"><table class="table" summary="Error rates for beta (incomplete)">
 488 <colgroup>
 489 <col>
 490 <col>
 491 <col>
 492 <col>
 493 <col>
 494 </colgroup>
 495 <thead><tr>
 496 <th>
 497               </th>
 498 <th>
 499                 <p>
 500                   GNU C++ version 7.1.0<br> linux<br> double
 501                 </p>
 502               </th>
 503 <th>
 504                 <p>
 505                   GNU C++ version 7.1.0<br> linux<br> long double
 506                 </p>
 507               </th>
 508 <th>
 509                 <p>
 510                   Sun compiler version 0x5150<br> Sun Solaris<br> long double
 511                 </p>
 512               </th>
 513 <th>
 514                 <p>
 515                   Microsoft Visual C++ version 14.1<br> Win32<br> double
 516                 </p>
 517               </th>
 518 </tr></thead>
 519 <tbody>
 520 <tr>
 521 <td>
 522                 <p>
 523                   Incomplete Beta Function: Small Values
 524                 </p>
 525               </td>
 526 <td>
 527                 <p>
 528                   <span class="blue">Max = 0&#949; (Mean = 0&#949;)</span>
 529                 </p>
 530               </td>
 531 <td>
 532                 <p>
 533                   <span class="blue">Max = 11.1&#949; (Mean = 2.32&#949;)</span>
 534                 </p>
 535               </td>
 536 <td>
 537                 <p>
 538                   <span class="blue">Max = 18.7&#949; (Mean = 3.19&#949;)</span>
 539                 </p>
 540               </td>
 541 <td>
 542                 <p>
 543                   <span class="blue">Max = 9.94&#949; (Mean = 2.17&#949;)</span>
 544                 </p>
 545               </td>
 546 </tr>
 547 <tr>
 548 <td>
 549                 <p>
 550                   Incomplete Beta Function: Medium Values
 551                 </p>
 552               </td>
 553 <td>
 554                 <p>
 555                   <span class="blue">Max = 0.568&#949; (Mean = 0.0254&#949;)</span>
 556                 </p>
 557               </td>
 558 <td>
 559                 <p>
 560                   <span class="blue">Max = 69.2&#949; (Mean = 13.4&#949;)</span>
 561                 </p>
 562               </td>
 563 <td>
 564                 <p>
 565                   <span class="blue">Max = 174&#949; (Mean = 25&#949;)</span>
 566                 </p>
 567               </td>
 568 <td>
 569                 <p>
 570                   <span class="blue">Max = 90&#949; (Mean = 12.7&#949;)</span>
 571                 </p>
 572               </td>
 573 </tr>
 574 <tr>
 575 <td>
 576                 <p>
 577                   Incomplete Beta Function: Large and Diverse Values
 578                 </p>
 579               </td>
 580 <td>
 581                 <p>
 582                   <span class="blue">Max = 0.999&#949; (Mean = 0.0325&#949;)</span>
 583                 </p>
 584               </td>
 585 <td>
 586                 <p>
 587                   <span class="blue">Max = 6.84e+04&#949; (Mean = 2.76e+03&#949;)</span>
 588                 </p>
 589               </td>
 590 <td>
 591                 <p>
 592                   <span class="blue">Max = 6.86e+04&#949; (Mean = 2.79e+03&#949;)</span>
 593                 </p>
 594               </td>
 595 <td>
 596                 <p>
 597                   <span class="blue">Max = 633&#949; (Mean = 29.7&#949;)</span>
 598                 </p>
 599               </td>
 600 </tr>
 601 <tr>
 602 <td>
 603                 <p>
 604                   Incomplete Beta Function: Small Integer Values
 605                 </p>
 606               </td>
 607 <td>
 608                 <p>
 609                   <span class="blue">Max = 0.786&#949; (Mean = 0.0323&#949;)</span>
 610                 </p>
 611               </td>
 612 <td>
 613                 <p>
 614                   <span class="blue">Max = 11.6&#949; (Mean = 3.6&#949;)</span>
 615                 </p>
 616               </td>
 617 <td>
 618                 <p>
 619                   <span class="blue">Max = 51.8&#949; (Mean = 11&#949;)</span>
 620                 </p>
 621               </td>
 622 <td>
 623                 <p>
 624                   <span class="blue">Max = 26&#949; (Mean = 6.28&#949;)</span>
 625                 </p>
 626               </td>
 627 </tr>
 628 </tbody>
 629 </table></div>
 630 </div>
 631 <br class="table-break"><div class="table">
 632 <a name="math_toolkit.sf_beta.ibeta_function.table_betac"></a><p class="title"><b>Table&#160;8.21.&#160;Error rates for betac</b></p>
 633 <div class="table-contents"><table class="table" summary="Error rates for betac">
 634 <colgroup>
 635 <col>
 636 <col>
 637 <col>
 638 <col>
 639 <col>
 640 </colgroup>
 641 <thead><tr>
 642 <th>
 643               </th>
 644 <th>
 645                 <p>
 646                   GNU C++ version 7.1.0<br> linux<br> double
 647                 </p>
 648               </th>
 649 <th>
 650                 <p>
 651                   GNU C++ version 7.1.0<br> linux<br> long double
 652                 </p>
 653               </th>
 654 <th>
 655                 <p>
 656                   Sun compiler version 0x5150<br> Sun Solaris<br> long double
 657                 </p>
 658               </th>
 659 <th>
 660                 <p>
 661                   Microsoft Visual C++ version 14.1<br> Win32<br> double
 662                 </p>
 663               </th>
 664 </tr></thead>
 665 <tbody>
 666 <tr>
 667 <td>
 668                 <p>
 669                   Incomplete Beta Function: Small Values
 670                 </p>
 671               </td>
 672 <td>
 673                 <p>
 674                   <span class="blue">Max = 0.676&#949; (Mean = 0.0302&#949;)</span>
 675                 </p>
 676               </td>
 677 <td>
 678                 <p>
 679                   <span class="blue">Max = 9.92&#949; (Mean = 2.3&#949;)</span>
 680                 </p>
 681               </td>
 682 <td>
 683                 <p>
 684                   <span class="blue">Max = 11.2&#949; (Mean = 2.94&#949;)</span>
 685                 </p>
 686               </td>
 687 <td>
 688                 <p>
 689                   <span class="blue">Max = 8.94&#949; (Mean = 2.06&#949;)</span>
 690                 </p>
 691               </td>
 692 </tr>
 693 <tr>
 694 <td>
 695                 <p>
 696                   Incomplete Beta Function: Medium Values
 697                 </p>
 698               </td>
 699 <td>
 700                 <p>
 701                   <span class="blue">Max = 0.949&#949; (Mean = 0.098&#949;)</span>
 702                 </p>
 703               </td>
 704 <td>
 705                 <p>
 706                   <span class="blue">Max = 63.5&#949; (Mean = 13.5&#949;)</span>
 707                 </p>
 708               </td>
 709 <td>
 710                 <p>
 711                   <span class="blue">Max = 97.6&#949; (Mean = 24.3&#949;)</span>
 712                 </p>
 713               </td>
 714 <td>
 715                 <p>
 716                   <span class="blue">Max = 90.6&#949; (Mean = 14.8&#949;)</span>
 717                 </p>
 718               </td>
 719 </tr>
 720 <tr>
 721 <td>
 722                 <p>
 723                   Incomplete Beta Function: Large and Diverse Values
 724                 </p>
 725               </td>
 726 <td>
 727                 <p>
 728                   <span class="blue">Max = 1.12&#949; (Mean = 0.0458&#949;)</span>
 729                 </p>
 730               </td>
 731 <td>
 732                 <p>
 733                   <span class="blue">Max = 1.05e+05&#949; (Mean = 5.45e+03&#949;)</span>
 734                 </p>
 735               </td>
 736 <td>
 737                 <p>
 738                   <span class="blue">Max = 1.04e+05&#949; (Mean = 5.46e+03&#949;)</span>
 739                 </p>
 740               </td>
 741 <td>
 742                 <p>
 743                   <span class="blue">Max = 3.72e+03&#949; (Mean = 113&#949;)</span>
 744                 </p>
 745               </td>
 746 </tr>
 747 <tr>
 748 <td>
 749                 <p>
 750                   Incomplete Beta Function: Small Integer Values
 751                 </p>
 752               </td>
 753 <td>
 754                 <p>
 755                   <span class="blue">Max = 0.586&#949; (Mean = 0.0314&#949;)</span>
 756                 </p>
 757               </td>
 758 <td>
 759                 <p>
 760                   <span class="blue">Max = 11.1&#949; (Mean = 3.65&#949;)</span>
 761                 </p>
 762               </td>
 763 <td>
 764                 <p>
 765                   <span class="blue">Max = 103&#949; (Mean = 17.4&#949;)</span>
 766                 </p>
 767               </td>
 768 <td>
 769                 <p>
 770                   <span class="blue">Max = 26.2&#949; (Mean = 6.36&#949;)</span>
 771                 </p>
 772               </td>
 773 </tr>
 774 </tbody>
 775 </table></div>
 776 </div>
 777 <br class="table-break"><h5>
 778 <a name="math_toolkit.sf_beta.ibeta_function.h3"></a>
 779         <span class="phrase"><a name="math_toolkit.sf_beta.ibeta_function.testing"></a></span><a class="link" href="ibeta_function.html#math_toolkit.sf_beta.ibeta_function.testing">Testing</a>
 780       </h5>
 781 <p>
 782         There are two sets of tests: spot tests compare values taken from <a href="http://functions.wolfram.com/webMathematica/FunctionEvaluation.jsp?name=BetaRegularized" target="_top">Mathworld's
 783         online function evaluator</a> with this implementation: they provide
 784         a basic "sanity check" for the implementation, with one spot-test
 785         in each implementation-domain (see implementation notes below).
 786       </p>
 787 <p>
 788         Accuracy tests use data generated at very high precision (with <a href="http://shoup.net/ntl/doc/RR.txt" target="_top">NTL
 789         RR class</a> set at 1000-bit precision), using the "textbook"
 790         continued fraction representation (refer to the first continued fraction
 791         in the implementation discussion below). Note that this continued fraction
 792         is <span class="emphasis"><em>not</em></span> used in the implementation, and therefore we
 793         have test data that is fully independent of the code.
 794       </p>
 795 <h5>
 796 <a name="math_toolkit.sf_beta.ibeta_function.h4"></a>
 797         <span class="phrase"><a name="math_toolkit.sf_beta.ibeta_function.implementation"></a></span><a class="link" href="ibeta_function.html#math_toolkit.sf_beta.ibeta_function.implementation">Implementation</a>
 798       </h5>
 799 <p>
 800         This implementation is closely based upon <a href="http://portal.acm.org/citation.cfm?doid=131766.131776" target="_top">"Algorithm
 801         708; Significant digit computation of the incomplete beta function ratios",
 802         DiDonato and Morris, ACM, 1992.</a>
 803       </p>
 804 <p>
 805         All four of these functions share a common implementation: this is passed
 806         both x and y, and can return either p or q where these are related by:
 807       </p>
 808 <div class="blockquote"><blockquote class="blockquote"><p>
 809           <span class="inlinemediaobject"><img src="../../../equations/ibeta_inv5.svg"></span>
 810
 811         </p></blockquote></div>
 812 <p>
 813         so at any point we can swap a for b, x for y and p for q if this results
 814         in a more favourable position. Generally such swaps are performed so that
 815         we always compute a value less than 0.9: when required this can then be subtracted
 816         from 1 without undue cancellation error.
 817       </p>
 818 <p>
 819         The following continued fraction representation is found in many textbooks
 820         but is not used in this implementation - it's both slower and less accurate
 821         than the alternatives - however it is used to generate test data:
 822       </p>
 823 <div class="blockquote"><blockquote class="blockquote"><p>
 824           <span class="inlinemediaobject"><img src="../../../equations/ibeta5.svg"></span>
 825
 826         </p></blockquote></div>
 827 <p>
 828         The following continued fraction is due to <a href="http://portal.acm.org/citation.cfm?doid=131766.131776" target="_top">Didonato
 829         and Morris</a>, and is used in this implementation when a and b are both
 830         greater than 1:
 831       </p>
 832 <div class="blockquote"><blockquote class="blockquote"><p>
 833           <span class="inlinemediaobject"><img src="../../../equations/ibeta6.svg"></span>
 834
 835         </p></blockquote></div>
 836 <p>
 837         For smallish b and x then a series representation can be used:
 838       </p>
 839 <div class="blockquote"><blockquote class="blockquote"><p>
 840           <span class="inlinemediaobject"><img src="../../../equations/ibeta7.svg"></span>
 841
 842         </p></blockquote></div>
 843 <p>
 844         When b &lt;&lt; a then the transition from 0 to 1 occurs very close to x
 845         = 1 and some care has to be taken over the method of computation, in that
 846         case the following series representation is used:
 847       </p>
 848 <div class="blockquote"><blockquote class="blockquote"><p>
 849           <span class="inlinemediaobject"><img src="../../../equations/ibeta8.svg"></span>
 850
 851         </p></blockquote></div>
 852 <p>
 853         Where Q(a,x) is an <a href="http://functions.wolfram.com/GammaBetaErf/Gamma2/" target="_top">incomplete
 854         gamma function</a>. Note that this method relies on keeping a table of
 855         all the p<sub>n </sub> previously computed, which does limit the precision of the method,
 856         depending upon the size of the table used.
 857       </p>
 858 <p>
 859         When <span class="emphasis"><em>a</em></span> and <span class="emphasis"><em>b</em></span> are both small integers,
 860         then we can relate the incomplete beta to the binomial distribution and use
 861         the following finite sum:
 862       </p>
 863 <div class="blockquote"><blockquote class="blockquote"><p>
 864           <span class="inlinemediaobject"><img src="../../../equations/ibeta12.svg"></span>
 865
 866         </p></blockquote></div>
 867 <p>
 868         Finally we can sidestep difficult areas, or move to an area with a more efficient
 869         means of computation, by using the duplication formulae:
 870       </p>
 871 <div class="blockquote"><blockquote class="blockquote"><p>
 872           <span class="inlinemediaobject"><img src="../../../equations/ibeta10.svg"></span>
 873
 874         </p></blockquote></div>
 875 <div class="blockquote"><blockquote class="blockquote"><p>
 876           <span class="inlinemediaobject"><img src="../../../equations/ibeta11.svg"></span>
 877
 878         </p></blockquote></div>
 879 <p>
 880         The domains of a, b and x for which the various methods are used are identical
 881         to those described in the <a href="http://portal.acm.org/citation.cfm?doid=131766.131776" target="_top">Didonato
 882         and Morris TOMS 708 paper</a>.
 883       </p>
 884 </div>
 885 <table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr>
 886 <td align="left"></td>
 887 <td align="right"><div class="copyright-footer">Copyright &#169; 2006-2019 Nikhar
 888       Agrawal, Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos,
 889       Hubert Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Matthew Pulver, Johan
 890       R&#229;de, Gautam Sewani, Benjamin Sobotta, Nicholas Thompson, Thijs van den Berg,
 891       Daryle Walker and Xiaogang Zhang<p>
 892         Distributed under the Boost Software License, Version 1.0. (See accompanying
 893         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>)
 894       </p>
 895 </div></td>
 896 </tr></table>
 897 <hr>
 898 <div class="spirit-nav">
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