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4<title>Incomplete Beta Functions</title>
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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">&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;15.&#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;15.&#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;15.&#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;15.&#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;15.&#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;15.&#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;15.&#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;15.&#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;15.&#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;15.&#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;15.&#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;15.&#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<p>
112 <span class="inlinemediaobject"><img src="../../../equations/ibeta3.svg"></span>
113 </p>
114<p>
115 <span class="inlinemediaobject"><img src="../../../graphs/ibeta.svg" align="middle"></span>
116 </p>
117<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>
118<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>
119
120<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;15.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&gt;</span>
121<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;15.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&amp;);</span>
122</pre>
123<p>
124 Returns the normalised complement of the incomplete beta function of a, b
125 and x:
126 </p>
127<p>
128 <span class="inlinemediaobject"><img src="../../../equations/ibeta4.svg"></span>
129 </p>
130<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>
131<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>
132
133<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;15.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><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> <span class="keyword">const</span> <a class="link" href="../../policy.html" title="Chapter&#160;15.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&amp;);</span>
135</pre>
136<p>
137 Returns the full (non-normalised) incomplete beta function of a, b and x:
138 </p>
139<p>
140 <span class="inlinemediaobject"><img src="../../../equations/ibeta1.svg"></span>
141 </p>
142<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>
143<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>
144
145<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;15.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&gt;</span>
146<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;15.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&amp;);</span>
147</pre>
148<p>
149 Returns the full (non-normalised) complement of the incomplete beta function
150 of a, b and x:
151 </p>
152<p>
153 <span class="inlinemediaobject"><img src="../../../equations/ibeta2.svg"></span>
154 </p>
155<h5>
156<a name="math_toolkit.sf_beta.ibeta_function.h2"></a>
157 <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>
158 </h5>
159<p>
160 The following tables give peak and mean relative errors in over various domains
161 of a, b and x, along with comparisons to the <a href="http://www.gnu.org/software/gsl/" target="_top">GSL-1.9</a>
162 and <a href="http://www.netlib.org/cephes/" target="_top">Cephes</a> libraries.
163 Note that only results for the widest floating-point type on the system are
164 given as narrower types have <a class="link" href="../relative_error.html#math_toolkit.relative_error.zero_error">effectively
165 zero error</a>.
166 </p>
167<p>
168 Note that the results for 80 and 128-bit long doubles are noticeably higher
169 than for doubles: this is because the wider exponent range of these types
170 allow more extreme test cases to be tested. For example expected results
171 that are zero at double precision, may be finite but exceptionally small
172 with the wider exponent range of the long double types.
173 </p>
174<div class="table">
175<a name="math_toolkit.sf_beta.ibeta_function.table_ibeta"></a><p class="title"><b>Table&#160;6.18.&#160;Error rates for ibeta</b></p>
176<div class="table-contents"><table class="table" summary="Error rates for ibeta">
177<colgroup>
178<col>
179<col>
180<col>
181<col>
182<col>
183</colgroup>
184<thead><tr>
185<th>
186 </th>
187<th>
188 <p>
189 Microsoft Visual C++ version 12.0<br> Win32<br> double
190 </p>
191 </th>
192<th>
193 <p>
194 GNU C++ version 5.1.0<br> linux<br> double
195 </p>
196 </th>
197<th>
198 <p>
199 GNU C++ version 5.1.0<br> linux<br> long double
200 </p>
201 </th>
202<th>
203 <p>
204 Sun compiler version 0x5130<br> Sun Solaris<br> long double
205 </p>
206 </th>
207</tr></thead>
208<tbody>
209<tr>
210<td>
211 <p>
212 Incomplete Beta Function: Small Values
213 </p>
214 </td>
215<td>
216 <p>
217 <span class="blue">Max = 9.42&#949; (Mean = 2.24&#949;)</span>
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 1.16:</em></span> Max = 682&#949; (Mean = 32.6&#949;))<br> (<span class="emphasis"><em>Rmath
224 3.0.2:</em></span> Max = 22.9&#949; (Mean = 3.35&#949;))<br> (<span class="emphasis"><em>Cephes:</em></span>
225 Max = 22.7&#949; (Mean = 1.69&#949;))
226 </p>
227 </td>
228<td>
229 <p>
230 <span class="blue">Max = 6.27&#949; (Mean = 1.7&#949;)</span>
231 </p>
232 </td>
233<td>
234 <p>
235 <span class="blue">Max = 21.3&#949; (Mean = 2.74&#949;)</span>
236 </p>
237 </td>
238</tr>
239<tr>
240<td>
241 <p>
242 Incomplete Beta Function: Medium Values
243 </p>
244 </td>
245<td>
246 <p>
247 <span class="blue">Max = 108&#949; (Mean = 16.3&#949;)</span>
248 </p>
249 </td>
250<td>
251 <p>
252 <span class="blue">Max = 0&#949; (Mean = 0&#949;)</span><br> <br> (<span class="emphasis"><em>GSL
253 1.16:</em></span> Max = 690&#949; (Mean = 151&#949;))<br> (<span class="emphasis"><em>Rmath
254 3.0.2:</em></span> Max = 232&#949; (Mean = 27.9&#949;))<br> (<span class="emphasis"><em>Cephes:</em></span>
255 Max = 1.55e+03&#949; (Mean = 121&#949;))
256 </p>
257 </td>
258<td>
259 <p>
260 <span class="blue">Max = 51.4&#949; (Mean = 12&#949;)</span>
261 </p>
262 </td>
263<td>
264 <p>
265 <span class="blue">Max = 124&#949; (Mean = 18.4&#949;)</span>
266 </p>
267 </td>
268</tr>
269<tr>
270<td>
271 <p>
272 Incomplete Beta Function: Large and Diverse Values
273 </p>
274 </td>
275<td>
276 <p>
277 <span class="blue">Max = 1.19e+003&#949; (Mean = 59.5&#949;)</span>
278 </p>
279 </td>
280<td>
281 <p>
282 <span class="blue">Max = 1.26&#949; (Mean = 0.063&#949;)</span><br> <br>
283 (<span class="emphasis"><em>GSL 1.16:</em></span> Max = 3.9e+05&#949; (Mean = 1.82e+04&#949;)
284 <a class="link" href="../logs_and_tables/logs.html#errors_GNU_C_version_5_1_0_linux_double_ibeta_GSL_1_16_Incomplete_Beta_Function_Large_and_Diverse_Values">And
285 other failures.</a>)<br> (<span class="emphasis"><em>Rmath 3.0.2:</em></span>
286 Max = 574&#949; (Mean = 49.4&#949;))<br> (<span class="emphasis"><em>Cephes:</em></span> Max
287 = 5.92e+05&#949; (Mean = 2.17e+04&#949;))
288 </p>
289 </td>
290<td>
291 <p>
292 <span class="blue">Max = 1.96e+04&#949; (Mean = 997&#949;)</span>
293 </p>
294 </td>
295<td>
296 <p>
297 <span class="blue">Max = 4.98e+04&#949; (Mean = 2.07e+03&#949;)</span>
298 </p>
299 </td>
300</tr>
301<tr>
302<td>
303 <p>
304 Incomplete Beta Function: Small Integer Values
305 </p>
306 </td>
307<td>
308 <p>
309 <span class="blue">Max = 3.69&#949; (Mean = 0.765&#949;)</span>
310 </p>
311 </td>
312<td>
313 <p>
314 <span class="blue">Max = 0&#949; (Mean = 0&#949;)</span><br> <br> (<span class="emphasis"><em>GSL
315 1.16:</em></span> Max = 254&#949; (Mean = 50.9&#949;))<br> (<span class="emphasis"><em>Rmath
316 3.0.2:</em></span> Max = 62.2&#949; (Mean = 8.95&#949;))<br> (<span class="emphasis"><em>Cephes:</em></span>
317 Max = 7.96&#949; (Mean = 1.3&#949;))
318 </p>
319 </td>
320<td>
321 <p>
322 <span class="blue">Max = 4.45&#949; (Mean = 0.814&#949;)</span>
323 </p>
324 </td>
325<td>
326 <p>
327 <span class="blue">Max = 44.5&#949; (Mean = 10.1&#949;)</span>
328 </p>
329 </td>
330</tr>
331</tbody>
332</table></div>
333</div>
334<br class="table-break"><div class="table">
335<a name="math_toolkit.sf_beta.ibeta_function.table_ibetac"></a><p class="title"><b>Table&#160;6.19.&#160;Error rates for ibetac</b></p>
336<div class="table-contents"><table class="table" summary="Error rates for ibetac">
337<colgroup>
338<col>
339<col>
340<col>
341<col>
342<col>
343</colgroup>
344<thead><tr>
345<th>
346 </th>
347<th>
348 <p>
349 Microsoft Visual C++ version 12.0<br> Win32<br> double
350 </p>
351 </th>
352<th>
353 <p>
354 GNU C++ version 5.1.0<br> linux<br> double
355 </p>
356 </th>
357<th>
358 <p>
359 GNU C++ version 5.1.0<br> linux<br> long double
360 </p>
361 </th>
362<th>
363 <p>
364 Sun compiler version 0x5130<br> Sun Solaris<br> long double
365 </p>
366 </th>
367</tr></thead>
368<tbody>
369<tr>
370<td>
371 <p>
372 Incomplete Beta Function: Small Values
373 </p>
374 </td>
375<td>
376 <p>
377 <span class="blue">Max = 6.56&#949; (Mean = 1.88&#949;)</span>
378 </p>
379 </td>
380<td>
381 <p>
382 <span class="blue">Max = 0&#949; (Mean = 0&#949;)</span><br> <br> (<span class="emphasis"><em>Rmath
383 3.0.2:</em></span> Max = 22.4&#949; (Mean = 3.67&#949;))
384 </p>
385 </td>
386<td>
387 <p>
388 <span class="blue">Max = 8.47&#949; (Mean = 1.68&#949;)</span>
389 </p>
390 </td>
391<td>
392 <p>
393 <span class="blue">Max = 13.8&#949; (Mean = 2.69&#949;)</span>
394 </p>
395 </td>
396</tr>
397<tr>
398<td>
399 <p>
400 Incomplete Beta Function: Medium Values
401 </p>
402 </td>
403<td>
404 <p>
405 <span class="blue">Max = 56.2&#949; (Mean = 14.1&#949;)</span>
406 </p>
407 </td>
408<td>
409 <p>
410 <span class="blue">Max = 0&#949; (Mean = 0&#949;)</span><br> <br> (<span class="emphasis"><em>Rmath
411 3.0.2:</em></span> Max = 204&#949; (Mean = 25.8&#949;))
412 </p>
413 </td>
414<td>
415 <p>
416 <span class="blue">Max = 75.3&#949; (Mean = 11.3&#949;)</span>
417 </p>
418 </td>
419<td>
420 <p>
421 <span class="blue">Max = 132&#949; (Mean = 19.8&#949;)</span>
422 </p>
423 </td>
424</tr>
425<tr>
426<td>
427 <p>
428 Incomplete Beta Function: Large and Diverse Values
429 </p>
430 </td>
431<td>
432 <p>
433 <span class="blue">Max = 1.87e+003&#949; (Mean = 93.8&#949;)</span>
434 </p>
435 </td>
436<td>
437 <p>
438 <span class="blue">Max = 0.981&#949; (Mean = 0.0638&#949;)</span><br>
439 <br> (<span class="emphasis"><em>Rmath 3.0.2:</em></span> Max = 889&#949; (Mean = 68.4&#949;))
440 </p>
441 </td>
442<td>
443 <p>
444 <span class="blue">Max = 3.45e+04&#949; (Mean = 1.32e+03&#949;)</span>
445 </p>
446 </td>
447<td>
448 <p>
449 <span class="blue">Max = 4.26e+04&#949; (Mean = 1.54e+03&#949;)</span>
450 </p>
451 </td>
452</tr>
453<tr>
454<td>
455 <p>
456 Incomplete Beta Function: Small Integer Values
457 </p>
458 </td>
459<td>
460 <p>
461 <span class="blue">Max = 6.37&#949; (Mean = 1.03&#949;)</span>
462 </p>
463 </td>
464<td>
465 <p>
466 <span class="blue">Max = 0&#949; (Mean = 0&#949;)</span><br> <br> (<span class="emphasis"><em>Rmath
467 3.0.2:</em></span> Max = 84.6&#949; (Mean = 18&#949;))
468 </p>
469 </td>
470<td>
471 <p>
472 <span class="blue">Max = 5.34&#949; (Mean = 1.11&#949;)</span>
473 </p>
474 </td>
475<td>
476 <p>
477 <span class="blue">Max = 107&#949; (Mean = 17.1&#949;)</span>
478 </p>
479 </td>
480</tr>
481</tbody>
482</table></div>
483</div>
484<br class="table-break"><div class="table">
485<a name="math_toolkit.sf_beta.ibeta_function.table_beta_incomplete_"></a><p class="title"><b>Table&#160;6.20.&#160;Error rates for beta (incomplete)</b></p>
486<div class="table-contents"><table class="table" summary="Error rates for beta (incomplete)">
487<colgroup>
488<col>
489<col>
490<col>
491<col>
492<col>
493</colgroup>
494<thead><tr>
495<th>
496 </th>
497<th>
498 <p>
499 Microsoft Visual C++ version 12.0<br> Win32<br> double
500 </p>
501 </th>
502<th>
503 <p>
504 GNU C++ version 5.1.0<br> linux<br> double
505 </p>
506 </th>
507<th>
508 <p>
509 GNU C++ version 5.1.0<br> linux<br> long double
510 </p>
511 </th>
512<th>
513 <p>
514 Sun compiler version 0x5130<br> Sun Solaris<br> long double
515 </p>
516 </th>
517</tr></thead>
518<tbody>
519<tr>
520<td>
521 <p>
522 Incomplete Beta Function: Small Values
523 </p>
524 </td>
525<td>
526 <p>
527 <span class="blue">Max = 11.1&#949; (Mean = 2.28&#949;)</span>
528 </p>
529 </td>
530<td>
531 <p>
532 <span class="blue">Max = 0&#949; (Mean = 0&#949;)</span>
533 </p>
534 </td>
535<td>
536 <p>
537 <span class="blue">Max = 12.5&#949; (Mean = 2.01&#949;)</span>
538 </p>
539 </td>
540<td>
541 <p>
542 <span class="blue">Max = 18.7&#949; (Mean = 3.19&#949;)</span>
543 </p>
544 </td>
545</tr>
546<tr>
547<td>
548 <p>
549 Incomplete Beta Function: Medium Values
550 </p>
551 </td>
552<td>
553 <p>
554 <span class="blue">Max = 91.1&#949; (Mean = 12.7&#949;)</span>
555 </p>
556 </td>
557<td>
558 <p>
559 <span class="blue">Max = 0.568&#949; (Mean = 0.0342&#949;)</span>
560 </p>
561 </td>
562<td>
563 <p>
564 <span class="blue">Max = 67.1&#949; (Mean = 13.3&#949;)</span>
565 </p>
566 </td>
567<td>
568 <p>
569 <span class="blue">Max = 174&#949; (Mean = 25&#949;)</span>
570 </p>
571 </td>
572</tr>
573<tr>
574<td>
575 <p>
576 Incomplete Beta Function: Large and Diverse Values
577 </p>
578 </td>
579<td>
580 <p>
581 <span class="blue">Max = 635&#949; (Mean = 29.5&#949;)</span>
582 </p>
583 </td>
584<td>
585 <p>
586 <span class="blue">Max = 0.999&#949; (Mean = 0.0399&#949;)</span>
587 </p>
588 </td>
589<td>
590 <p>
591 <span class="blue">Max = 6.84e+04&#949; (Mean = 2.76e+03&#949;)</span>
592 </p>
593 </td>
594<td>
595 <p>
596 <span class="blue">Max = 6.86e+04&#949; (Mean = 2.79e+03&#949;)</span>
597 </p>
598 </td>
599</tr>
600<tr>
601<td>
602 <p>
603 Incomplete Beta Function: Small Integer Values
604 </p>
605 </td>
606<td>
607 <p>
608 <span class="blue">Max = 26.8&#949; (Mean = 6.61&#949;)</span>
609 </p>
610 </td>
611<td>
612 <p>
613 <span class="blue">Max = 0.786&#949; (Mean = 0.0248&#949;)</span>
614 </p>
615 </td>
616<td>
617 <p>
618 <span class="blue">Max = 10.2&#949; (Mean = 3.49&#949;)</span>
619 </p>
620 </td>
621<td>
622 <p>
623 <span class="blue">Max = 51.8&#949; (Mean = 11&#949;)</span>
624 </p>
625 </td>
626</tr>
627</tbody>
628</table></div>
629</div>
630<br class="table-break"><div class="table">
631<a name="math_toolkit.sf_beta.ibeta_function.table_betac"></a><p class="title"><b>Table&#160;6.21.&#160;Error rates for betac</b></p>
632<div class="table-contents"><table class="table" summary="Error rates for betac">
633<colgroup>
634<col>
635<col>
636<col>
637<col>
638<col>
639</colgroup>
640<thead><tr>
641<th>
642 </th>
643<th>
644 <p>
645 Microsoft Visual C++ version 12.0<br> Win32<br> double
646 </p>
647 </th>
648<th>
649 <p>
650 GNU C++ version 5.1.0<br> linux<br> double
651 </p>
652 </th>
653<th>
654 <p>
655 GNU C++ version 5.1.0<br> linux<br> long double
656 </p>
657 </th>
658<th>
659 <p>
660 Sun compiler version 0x5130<br> Sun Solaris<br> long double
661 </p>
662 </th>
663</tr></thead>
664<tbody>
665<tr>
666<td>
667 <p>
668 Incomplete Beta Function: Small Values
669 </p>
670 </td>
671<td>
672 <p>
673 <span class="blue">Max = 12&#949; (Mean = 2.43&#949;)</span>
674 </p>
675 </td>
676<td>
677 <p>
678 <span class="blue">Max = 0.676&#949; (Mean = 0.0302&#949;)</span>
679 </p>
680 </td>
681<td>
682 <p>
683 <span class="blue">Max = 7.75&#949; (Mean = 1.91&#949;)</span>
684 </p>
685 </td>
686<td>
687 <p>
688 <span class="blue">Max = 12.6&#949; (Mean = 3.02&#949;)</span>
689 </p>
690 </td>
691</tr>
692<tr>
693<td>
694 <p>
695 Incomplete Beta Function: Medium Values
696 </p>
697 </td>
698<td>
699 <p>
700 <span class="blue">Max = 91.3&#949; (Mean = 14.8&#949;)</span>
701 </p>
702 </td>
703<td>
704 <p>
705 <span class="blue">Max = 0.949&#949; (Mean = 0.0944&#949;)</span>
706 </p>
707 </td>
708<td>
709 <p>
710 <span class="blue">Max = 63.5&#949; (Mean = 13.4&#949;)</span>
711 </p>
712 </td>
713<td>
714 <p>
715 <span class="blue">Max = 97.6&#949; (Mean = 24.3&#949;)</span>
716 </p>
717 </td>
718</tr>
719<tr>
720<td>
721 <p>
722 Incomplete Beta Function: Large and Diverse Values
723 </p>
724 </td>
725<td>
726 <p>
727 <span class="blue">Max = 3.72e+003&#949; (Mean = 113&#949;)</span>
728 </p>
729 </td>
730<td>
731 <p>
732 <span class="blue">Max = 1.12&#949; (Mean = 0.0458&#949;)</span>
733 </p>
734 </td>
735<td>
736 <p>
737 <span class="blue">Max = 1.05e+05&#949; (Mean = 5.45e+03&#949;)</span>
738 </p>
739 </td>
740<td>
741 <p>
742 <span class="blue">Max = 1.04e+05&#949; (Mean = 5.46e+03&#949;)</span>
743 </p>
744 </td>
745</tr>
746<tr>
747<td>
748 <p>
749 Incomplete Beta Function: Small Integer Values
750 </p>
751 </td>
752<td>
753 <p>
754 <span class="blue">Max = 26.7&#949; (Mean = 6.67&#949;)</span>
755 </p>
756 </td>
757<td>
758 <p>
759 <span class="blue">Max = 0.586&#949; (Mean = 0.0314&#949;)</span>
760 </p>
761 </td>
762<td>
763 <p>
764 <span class="blue">Max = 10.4&#949; (Mean = 3.54&#949;)</span>
765 </p>
766 </td>
767<td>
768 <p>
769 <span class="blue">Max = 103&#949; (Mean = 17.4&#949;)</span>
770 </p>
771 </td>
772</tr>
773</tbody>
774</table></div>
775</div>
776<br class="table-break"><h5>
777<a name="math_toolkit.sf_beta.ibeta_function.h3"></a>
778 <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>
779 </h5>
780<p>
781 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
782 online function evaluator</a> with this implementation: they provide
783 a basic "sanity check" for the implementation, with one spot-test
784 in each implementation-domain (see implementation notes below).
785 </p>
786<p>
787 Accuracy tests use data generated at very high precision (with <a href="http://shoup.net/ntl/doc/RR.txt" target="_top">NTL
788 RR class</a> set at 1000-bit precision), using the "textbook"
789 continued fraction representation (refer to the first continued fraction
790 in the implementation discussion below). Note that this continued fraction
791 is <span class="emphasis"><em>not</em></span> used in the implementation, and therefore we
792 have test data that is fully independent of the code.
793 </p>
794<h5>
795<a name="math_toolkit.sf_beta.ibeta_function.h4"></a>
796 <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>
797 </h5>
798<p>
799 This implementation is closely based upon <a href="http://portal.acm.org/citation.cfm?doid=131766.131776" target="_top">"Algorithm
800 708; Significant digit computation of the incomplete beta function ratios",
801 DiDonato and Morris, ACM, 1992.</a>
802 </p>
803<p>
804 All four of these functions share a common implementation: this is passed
805 both x and y, and can return either p or q where these are related by:
806 </p>
807<p>
808 <span class="inlinemediaobject"><img src="../../../equations/ibeta_inv5.svg"></span>
809 </p>
810<p>
811 so at any point we can swap a for b, x for y and p for q if this results
812 in a more favourable position. Generally such swaps are performed so that
813 we always compute a value less than 0.9: when required this can then be subtracted
814 from 1 without undue cancellation error.
815 </p>
816<p>
817 The following continued fraction representation is found in many textbooks
818 but is not used in this implementation - it's both slower and less accurate
819 than the alternatives - however it is used to generate test data:
820 </p>
821<p>
822 <span class="inlinemediaobject"><img src="../../../equations/ibeta5.svg"></span>
823 </p>
824<p>
825 The following continued fraction is due to <a href="http://portal.acm.org/citation.cfm?doid=131766.131776" target="_top">Didonato
826 and Morris</a>, and is used in this implementation when a and b are both
827 greater than 1:
828 </p>
829<p>
830 <span class="inlinemediaobject"><img src="../../../equations/ibeta6.svg"></span>
831 </p>
832<p>
833 For smallish b and x then a series representation can be used:
834 </p>
835<p>
836 <span class="inlinemediaobject"><img src="../../../equations/ibeta7.svg"></span>
837 </p>
838<p>
839 When b &lt;&lt; a then the transition from 0 to 1 occurs very close to x
840 = 1 and some care has to be taken over the method of computation, in that
841 case the following series representation is used:
842 </p>
843<p>
844 <span class="inlinemediaobject"><img src="../../../equations/ibeta8.svg"></span>
845
846 </p>
847<p>
848 Where Q(a,x) is an <a href="http://functions.wolfram.com/GammaBetaErf/Gamma2/" target="_top">incomplete
849 gamma function</a>. Note that this method relies on keeping a table of
850 all the p<sub>n </sub> previously computed, which does limit the precision of the method,
851 depending upon the size of the table used.
852 </p>
853<p>
854 When <span class="emphasis"><em>a</em></span> and <span class="emphasis"><em>b</em></span> are both small integers,
855 then we can relate the incomplete beta to the binomial distribution and use
856 the following finite sum:
857 </p>
858<p>
859 <span class="inlinemediaobject"><img src="../../../equations/ibeta12.svg"></span>
860 </p>
861<p>
862 Finally we can sidestep difficult areas, or move to an area with a more efficient
863 means of computation, by using the duplication formulae:
864 </p>
865<p>
866 <span class="inlinemediaobject"><img src="../../../equations/ibeta10.svg"></span>
867 </p>
868<p>
869 <span class="inlinemediaobject"><img src="../../../equations/ibeta11.svg"></span>
870 </p>
871<p>
872 The domains of a, b and x for which the various methods are used are identical
873 to those described in the <a href="http://portal.acm.org/citation.cfm?doid=131766.131776" target="_top">Didonato
874 and Morris TOMS 708 paper</a>.
875 </p>
876</div>
877<table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr>
878<td align="left"></td>
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880 Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos, Hubert
881 Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Johan R&#229;de, Gautam Sewani,
882 Benjamin Sobotta, Thijs van den Berg, Daryle Walker and Xiaogang Zhang<p>
883 Distributed under the Boost Software License, Version 1.0. (See accompanying
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