1<html> 2<head> 3<meta http-equiv="Content-Type" content="text/html; charset=UTF-8"> 4<title>Incomplete Beta Functions</title> 5<link rel="stylesheet" href="../../math.css" type="text/css"> 6<meta name="generator" content="DocBook XSL Stylesheets V1.79.1"> 7<link rel="home" href="../../index.html" title="Math Toolkit 2.12.0"> 8<link rel="up" href="../sf_beta.html" title="Beta Functions"> 9<link rel="prev" href="beta_function.html" title="Beta"> 10<link rel="next" href="ibeta_inv_function.html" title="The Incomplete Beta Function Inverses"> 11</head> 12<body bgcolor="white" text="black" link="#0000FF" vlink="#840084" alink="#0000FF"> 13<table cellpadding="2" width="100%"><tr> 14<td valign="top"><img alt="Boost C++ Libraries" width="277" height="86" src="../../../../../../boost.png"></td> 15<td align="center"><a href="../../../../../../index.html">Home</a></td> 16<td align="center"><a href="../../../../../../libs/libraries.htm">Libraries</a></td> 17<td align="center"><a href="http://www.boost.org/users/people.html">People</a></td> 18<td align="center"><a href="http://www.boost.org/users/faq.html">FAQ</a></td> 19<td align="center"><a href="../../../../../../more/index.htm">More</a></td> 20</tr></table> 21<hr> 22<div class="spirit-nav"> 23<a accesskey="p" href="beta_function.html"><img src="../../../../../../doc/src/images/prev.png" alt="Prev"></a><a accesskey="u" href="../sf_beta.html"><img src="../../../../../../doc/src/images/up.png" alt="Up"></a><a accesskey="h" href="../../index.html"><img src="../../../../../../doc/src/images/home.png" alt="Home"></a><a accesskey="n" href="ibeta_inv_function.html"><img src="../../../../../../doc/src/images/next.png" alt="Next"></a> 24</div> 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"><</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> 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"><</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> 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"><</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 21. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">></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 21. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&);</span> 43 44<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> <span class="identifier">T3</span><span class="special">></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"><</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 21. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">></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 21. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&);</span> 49 50<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> <span class="identifier">T3</span><span class="special">></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"><</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 21. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">></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 21. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&);</span> 55 56<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> <span class="identifier">T3</span><span class="special">></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"><</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 21. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">></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 21. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&);</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 <= x <= 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 >= 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 > 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 21. 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 21. 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"><</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> 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"><</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 21. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">></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 21. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&);</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"><</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> 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"><</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 21. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">></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 21. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&);</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"><</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> 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"><</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 21. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">></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 21. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&);</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"><</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> 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"><</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 21. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">></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 21. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&);</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 8.18. 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 = 0ε (Mean = 0ε)</span><br> <br> (<span class="emphasis"><em>GSL 223 2.1:</em></span> Max = 682ε (Mean = 32.6ε))<br> (<span class="emphasis"><em>Rmath 224 3.2.3:</em></span> Max = 22.9ε (Mean = 3.35ε)) 225 </p> 226 </td> 227<td> 228 <p> 229 <span class="blue">Max = 8.97ε (Mean = 2.09ε)</span> 230 </p> 231 </td> 232<td> 233 <p> 234 <span class="blue">Max = 21.3ε (Mean = 2.75ε)</span> 235 </p> 236 </td> 237<td> 238 <p> 239 <span class="blue">Max = 8.4ε (Mean = 1.93ε)</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 = 0ε (Mean = 0ε)</span><br> <br> (<span class="emphasis"><em>GSL 252 2.1:</em></span> Max = 690ε (Mean = 151ε))<br> (<span class="emphasis"><em>Rmath 3.2.3:</em></span> 253 Max = 232ε (Mean = 27.9ε)) 254 </p> 255 </td> 256<td> 257 <p> 258 <span class="blue">Max = 50ε (Mean = 12.1ε)</span> 259 </p> 260 </td> 261<td> 262 <p> 263 <span class="blue">Max = 124ε (Mean = 18.4ε)</span> 264 </p> 265 </td> 266<td> 267 <p> 268 <span class="blue">Max = 106ε (Mean = 16.3ε)</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 = 1.26ε (Mean = 0.063ε)</span><br> <br> 281 (<span class="emphasis"><em>GSL 2.1:</em></span> Max = 3.9e+05ε (Mean = 1.82e+04ε) 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 = 574ε (Mean = 49.4ε)) 285 </p> 286 </td> 287<td> 288 <p> 289 <span class="blue">Max = 1.96e+04ε (Mean = 997ε)</span> 290 </p> 291 </td> 292<td> 293 <p> 294 <span class="blue">Max = 4.98e+04ε (Mean = 2.07e+03ε)</span> 295 </p> 296 </td> 297<td> 298 <p> 299 <span class="blue">Max = 1.32e+03ε (Mean = 68.5ε)</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 = 0ε (Mean = 0ε)</span><br> <br> (<span class="emphasis"><em>GSL 312 2.1:</em></span> Max = 254ε (Mean = 50.9ε))<br> (<span class="emphasis"><em>Rmath 313 3.2.3:</em></span> Max = 62.2ε (Mean = 8.95ε)) 314 </p> 315 </td> 316<td> 317 <p> 318 <span class="blue">Max = 4.45ε (Mean = 0.814ε)</span> 319 </p> 320 </td> 321<td> 322 <p> 323 <span class="blue">Max = 44.5ε (Mean = 10.1ε)</span> 324 </p> 325 </td> 326<td> 327 <p> 328 <span class="blue">Max = 3.85ε (Mean = 0.791ε)</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 8.19. 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 = 0ε (Mean = 0ε)</span><br> <br> (<span class="emphasis"><em>Rmath 379 3.2.3:</em></span> Max = 22.4ε (Mean = 3.67ε)) 380 </p> 381 </td> 382<td> 383 <p> 384 <span class="blue">Max = 10.6ε (Mean = 2.22ε)</span> 385 </p> 386 </td> 387<td> 388 <p> 389 <span class="blue">Max = 13.8ε (Mean = 2.68ε)</span> 390 </p> 391 </td> 392<td> 393 <p> 394 <span class="blue">Max = 6.94ε (Mean = 1.71ε)</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 = 0ε (Mean = 0ε)</span><br> <br> (<span class="emphasis"><em>Rmath 407 3.2.3:</em></span> Max = 204ε (Mean = 25.8ε)) 408 </p> 409 </td> 410<td> 411 <p> 412 <span class="blue">Max = 73.9ε (Mean = 11.2ε)</span> 413 </p> 414 </td> 415<td> 416 <p> 417 <span class="blue">Max = 132ε (Mean = 19.8ε)</span> 418 </p> 419 </td> 420<td> 421 <p> 422 <span class="blue">Max = 56.7ε (Mean = 14.3ε)</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 = 0.981ε (Mean = 0.0573ε)</span><br> 435 <br> (<span class="emphasis"><em>Rmath 3.2.3:</em></span> Max = 889ε (Mean = 68.4ε)) 436 </p> 437 </td> 438<td> 439 <p> 440 <span class="blue">Max = 3.45e+04ε (Mean = 1.32e+03ε)</span> 441 </p> 442 </td> 443<td> 444 <p> 445 <span class="blue">Max = 6.31e+04ε (Mean = 2.04e+03ε)</span> 446 </p> 447 </td> 448<td> 449 <p> 450 <span class="blue">Max = 1.88e+03ε (Mean = 82.7ε)</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 = 0ε (Mean = 0ε)</span><br> <br> (<span class="emphasis"><em>Rmath 463 3.2.3:</em></span> Max = 84.6ε (Mean = 18ε)) 464 </p> 465 </td> 466<td> 467 <p> 468 <span class="blue">Max = 5.34ε (Mean = 1.11ε)</span> 469 </p> 470 </td> 471<td> 472 <p> 473 <span class="blue">Max = 107ε (Mean = 17.1ε)</span> 474 </p> 475 </td> 476<td> 477 <p> 478 <span class="blue">Max = 6.37ε (Mean = 1.03ε)</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 8.20. 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 = 0ε (Mean = 0ε)</span> 529 </p> 530 </td> 531<td> 532 <p> 533 <span class="blue">Max = 11.1ε (Mean = 2.32ε)</span> 534 </p> 535 </td> 536<td> 537 <p> 538 <span class="blue">Max = 18.7ε (Mean = 3.19ε)</span> 539 </p> 540 </td> 541<td> 542 <p> 543 <span class="blue">Max = 9.94ε (Mean = 2.17ε)</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 = 0.568ε (Mean = 0.0254ε)</span> 556 </p> 557 </td> 558<td> 559 <p> 560 <span class="blue">Max = 69.2ε (Mean = 13.4ε)</span> 561 </p> 562 </td> 563<td> 564 <p> 565 <span class="blue">Max = 174ε (Mean = 25ε)</span> 566 </p> 567 </td> 568<td> 569 <p> 570 <span class="blue">Max = 90ε (Mean = 12.7ε)</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 = 0.999ε (Mean = 0.0325ε)</span> 583 </p> 584 </td> 585<td> 586 <p> 587 <span class="blue">Max = 6.84e+04ε (Mean = 2.76e+03ε)</span> 588 </p> 589 </td> 590<td> 591 <p> 592 <span class="blue">Max = 6.86e+04ε (Mean = 2.79e+03ε)</span> 593 </p> 594 </td> 595<td> 596 <p> 597 <span class="blue">Max = 633ε (Mean = 29.7ε)</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 = 0.786ε (Mean = 0.0323ε)</span> 610 </p> 611 </td> 612<td> 613 <p> 614 <span class="blue">Max = 11.6ε (Mean = 3.6ε)</span> 615 </p> 616 </td> 617<td> 618 <p> 619 <span class="blue">Max = 51.8ε (Mean = 11ε)</span> 620 </p> 621 </td> 622<td> 623 <p> 624 <span class="blue">Max = 26ε (Mean = 6.28ε)</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 8.21. 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 = 0.676ε (Mean = 0.0302ε)</span> 675 </p> 676 </td> 677<td> 678 <p> 679 <span class="blue">Max = 9.92ε (Mean = 2.3ε)</span> 680 </p> 681 </td> 682<td> 683 <p> 684 <span class="blue">Max = 11.2ε (Mean = 2.94ε)</span> 685 </p> 686 </td> 687<td> 688 <p> 689 <span class="blue">Max = 8.94ε (Mean = 2.06ε)</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 = 0.949ε (Mean = 0.098ε)</span> 702 </p> 703 </td> 704<td> 705 <p> 706 <span class="blue">Max = 63.5ε (Mean = 13.5ε)</span> 707 </p> 708 </td> 709<td> 710 <p> 711 <span class="blue">Max = 97.6ε (Mean = 24.3ε)</span> 712 </p> 713 </td> 714<td> 715 <p> 716 <span class="blue">Max = 90.6ε (Mean = 14.8ε)</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 = 1.12ε (Mean = 0.0458ε)</span> 729 </p> 730 </td> 731<td> 732 <p> 733 <span class="blue">Max = 1.05e+05ε (Mean = 5.45e+03ε)</span> 734 </p> 735 </td> 736<td> 737 <p> 738 <span class="blue">Max = 1.04e+05ε (Mean = 5.46e+03ε)</span> 739 </p> 740 </td> 741<td> 742 <p> 743 <span class="blue">Max = 3.72e+03ε (Mean = 113ε)</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 = 0.586ε (Mean = 0.0314ε)</span> 756 </p> 757 </td> 758<td> 759 <p> 760 <span class="blue">Max = 11.1ε (Mean = 3.65ε)</span> 761 </p> 762 </td> 763<td> 764 <p> 765 <span class="blue">Max = 103ε (Mean = 17.4ε)</span> 766 </p> 767 </td> 768<td> 769 <p> 770 <span class="blue">Max = 26.2ε (Mean = 6.36ε)</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 << 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 © 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 Råde, 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"> 899<a accesskey="p" href="beta_function.html"><img src="../../../../../../doc/src/images/prev.png" alt="Prev"></a><a accesskey="u" href="../sf_beta.html"><img src="../../../../../../doc/src/images/up.png" alt="Up"></a><a accesskey="h" href="../../index.html"><img src="../../../../../../doc/src/images/home.png" alt="Home"></a><a accesskey="n" href="ibeta_inv_function.html"><img src="../../../../../../doc/src/images/next.png" alt="Next"></a> 900</div> 901</body> 902</html> 903
