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4 | <title>The Incomplete Beta Function Inverses</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_inv_function"></a><a class="link" href="ibeta_inv_function.html" title="The Incomplete Beta Function Inverses">The Incomplete | |
28 | Beta Function Inverses</a> | |
29 | </h3></div></div></div> | |
30 | <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> | |
31 | </pre> | |
32 | <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> | |
33 | ||
34 | <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> | |
35 | <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_inv</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">p</span><span class="special">);</span> | |
36 | ||
37 | <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 15. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">></span> | |
38 | <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_inv</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">p</span><span class="special">,</span> <span class="keyword">const</span> <a class="link" href="../../policy.html" title="Chapter 15. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&);</span> | |
39 | ||
40 | <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> <span class="identifier">T4</span><span class="special">></span> | |
41 | <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_inv</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">p</span><span class="special">,</span> <span class="identifier">T4</span><span class="special">*</span> <span class="identifier">py</span><span class="special">);</span> | |
42 | ||
43 | <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> <span class="identifier">T4</span><span class="special">,</span> <span class="keyword">class</span> <a class="link" href="../../policy.html" title="Chapter 15. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">></span> | |
44 | <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_inv</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">p</span><span class="special">,</span> <span class="identifier">T4</span><span class="special">*</span> <span class="identifier">py</span><span class="special">,</span> <span class="keyword">const</span> <a class="link" href="../../policy.html" title="Chapter 15. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&);</span> | |
45 | ||
46 | <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> | |
47 | <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_inv</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">q</span><span class="special">);</span> | |
48 | ||
49 | <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 15. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">></span> | |
50 | <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_inv</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">q</span><span class="special">,</span> <span class="keyword">const</span> <a class="link" href="../../policy.html" title="Chapter 15. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&);</span> | |
51 | ||
52 | <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> <span class="identifier">T4</span><span class="special">></span> | |
53 | <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_inv</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">q</span><span class="special">,</span> <span class="identifier">T4</span><span class="special">*</span> <span class="identifier">py</span><span class="special">);</span> | |
54 | ||
55 | <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> <span class="identifier">T4</span><span class="special">,</span> <span class="keyword">class</span> <a class="link" href="../../policy.html" title="Chapter 15. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">></span> | |
56 | <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_inv</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">q</span><span class="special">,</span> <span class="identifier">T4</span><span class="special">*</span> <span class="identifier">py</span><span class="special">,</span> <span class="keyword">const</span> <a class="link" href="../../policy.html" title="Chapter 15. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&);</span> | |
57 | ||
58 | <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> | |
59 | <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_inva</span><span class="special">(</span><span class="identifier">T1</span> <span class="identifier">b</span><span class="special">,</span> <span class="identifier">T2</span> <span class="identifier">x</span><span class="special">,</span> <span class="identifier">T3</span> <span class="identifier">p</span><span class="special">);</span> | |
60 | ||
61 | <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 15. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">></span> | |
62 | <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_inva</span><span class="special">(</span><span class="identifier">T1</span> <span class="identifier">b</span><span class="special">,</span> <span class="identifier">T2</span> <span class="identifier">x</span><span class="special">,</span> <span class="identifier">T3</span> <span class="identifier">p</span><span class="special">,</span> <span class="keyword">const</span> <a class="link" href="../../policy.html" title="Chapter 15. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&);</span> | |
63 | ||
64 | <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> | |
65 | <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_inva</span><span class="special">(</span><span class="identifier">T1</span> <span class="identifier">b</span><span class="special">,</span> <span class="identifier">T2</span> <span class="identifier">x</span><span class="special">,</span> <span class="identifier">T3</span> <span class="identifier">q</span><span class="special">);</span> | |
66 | ||
67 | <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 15. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">></span> | |
68 | <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_inva</span><span class="special">(</span><span class="identifier">T1</span> <span class="identifier">b</span><span class="special">,</span> <span class="identifier">T2</span> <span class="identifier">x</span><span class="special">,</span> <span class="identifier">T3</span> <span class="identifier">q</span><span class="special">,</span> <span class="keyword">const</span> <a class="link" href="../../policy.html" title="Chapter 15. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&);</span> | |
69 | ||
70 | <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> | |
71 | <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_invb</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">x</span><span class="special">,</span> <span class="identifier">T3</span> <span class="identifier">p</span><span class="special">);</span> | |
72 | ||
73 | <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 15. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">></span> | |
74 | <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_invb</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">x</span><span class="special">,</span> <span class="identifier">T3</span> <span class="identifier">p</span><span class="special">,</span> <span class="keyword">const</span> <a class="link" href="../../policy.html" title="Chapter 15. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&);</span> | |
75 | ||
76 | <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> | |
77 | <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_invb</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">x</span><span class="special">,</span> <span class="identifier">T3</span> <span class="identifier">q</span><span class="special">);</span> | |
78 | ||
79 | <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 15. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">></span> | |
80 | <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_invb</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">x</span><span class="special">,</span> <span class="identifier">T3</span> <span class="identifier">q</span><span class="special">,</span> <span class="keyword">const</span> <a class="link" href="../../policy.html" title="Chapter 15. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&);</span> | |
81 | ||
82 | <span class="special">}}</span> <span class="comment">// namespaces</span> | |
83 | </pre> | |
84 | <h5> | |
85 | <a name="math_toolkit.sf_beta.ibeta_inv_function.h0"></a> | |
86 | <span class="phrase"><a name="math_toolkit.sf_beta.ibeta_inv_function.description"></a></span><a class="link" href="ibeta_inv_function.html#math_toolkit.sf_beta.ibeta_inv_function.description">Description</a> | |
87 | </h5> | |
88 | <p> | |
89 | There are six <a href="http://functions.wolfram.com/GammaBetaErf/" target="_top">incomplete | |
90 | beta function inverses</a> which allow you solve for any of the three | |
91 | parameters to the incomplete beta, starting from either the result of the | |
92 | incomplete beta (p) or its complement (q). | |
93 | </p> | |
94 | <p> | |
95 | The final <a class="link" href="../../policy.html" title="Chapter 15. Policies: Controlling Precision, Error Handling etc">Policy</a> argument is optional and can | |
96 | be used to control the behaviour of the function: how it handles errors, | |
97 | what level of precision to use etc. Refer to the <a class="link" href="../../policy.html" title="Chapter 15. Policies: Controlling Precision, Error Handling etc">policy | |
98 | documentation for more details</a>. | |
99 | </p> | |
100 | <div class="tip"><table border="0" summary="Tip"> | |
101 | <tr> | |
102 | <td rowspan="2" align="center" valign="top" width="25"><img alt="[Tip]" src="../../../../../../doc/src/images/tip.png"></td> | |
103 | <th align="left">Tip</th> | |
104 | </tr> | |
105 | <tr><td align="left" valign="top"> | |
106 | <p> | |
107 | When people normally talk about the inverse of the incomplete beta function, | |
108 | they are talking about inverting on parameter <span class="emphasis"><em>x</em></span>. These | |
109 | are implemented here as ibeta_inv and ibetac_inv, and are by far the most | |
110 | efficient of the inverses presented here. | |
111 | </p> | |
112 | <p> | |
113 | The inverses on the <span class="emphasis"><em>a</em></span> and <span class="emphasis"><em>b</em></span> parameters | |
114 | find use in some statistical applications, but have to be computed by rather | |
115 | brute force numerical techniques and are consequently several times slower. | |
116 | These are implemented here as ibeta_inva and ibeta_invb, and complement | |
117 | versions ibetac_inva and ibetac_invb. | |
118 | </p> | |
119 | </td></tr> | |
120 | </table></div> | |
121 | <p> | |
122 | 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 | |
123 | type calculation rules</em></span></a> when called with arguments T1...TN | |
124 | of different types. | |
125 | </p> | |
126 | <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> | |
127 | <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_inv</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">p</span><span class="special">);</span> | |
128 | ||
129 | <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 15. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">></span> | |
130 | <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_inv</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">p</span><span class="special">,</span> <span class="keyword">const</span> <a class="link" href="../../policy.html" title="Chapter 15. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&);</span> | |
131 | ||
132 | <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> <span class="identifier">T4</span><span class="special">></span> | |
133 | <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_inv</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">p</span><span class="special">,</span> <span class="identifier">T4</span><span class="special">*</span> <span class="identifier">py</span><span class="special">);</span> | |
134 | ||
135 | <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> <span class="identifier">T4</span><span class="special">,</span> <span class="keyword">class</span> <a class="link" href="../../policy.html" title="Chapter 15. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">></span> | |
136 | <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_inv</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">p</span><span class="special">,</span> <span class="identifier">T4</span><span class="special">*</span> <span class="identifier">py</span><span class="special">,</span> <span class="keyword">const</span> <a class="link" href="../../policy.html" title="Chapter 15. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&);</span> | |
137 | </pre> | |
138 | <p> | |
139 | Returns a value <span class="emphasis"><em>x</em></span> such that: <code class="computeroutput"><span class="identifier">p</span> | |
140 | <span class="special">=</span> <span class="identifier">ibeta</span><span class="special">(</span><span class="identifier">a</span><span class="special">,</span> | |
141 | <span class="identifier">b</span><span class="special">,</span> <span class="identifier">x</span><span class="special">);</span></code> and | |
142 | sets <code class="computeroutput"><span class="special">*</span><span class="identifier">py</span> | |
143 | <span class="special">=</span> <span class="number">1</span> <span class="special">-</span> <span class="identifier">x</span></code> when | |
144 | the <code class="computeroutput"><span class="identifier">py</span></code> parameter is provided | |
145 | and is non-null. Note that internally this function computes whichever is | |
146 | the smaller of <code class="computeroutput"><span class="identifier">x</span></code> and <code class="computeroutput"><span class="number">1</span><span class="special">-</span><span class="identifier">x</span></code>, | |
147 | and therefore the value assigned to <code class="computeroutput"><span class="special">*</span><span class="identifier">py</span></code> is free from cancellation errors. That | |
148 | means that even if the function returns <code class="computeroutput"><span class="number">1</span></code>, | |
149 | the value stored in <code class="computeroutput"><span class="special">*</span><span class="identifier">py</span></code> | |
150 | may be non-zero, albeit very small. | |
151 | </p> | |
152 | <p> | |
153 | Requires: <span class="emphasis"><em>a,b > 0</em></span> and <span class="emphasis"><em>0 <= p <= 1</em></span>. | |
154 | </p> | |
155 | <p> | |
156 | The final <a class="link" href="../../policy.html" title="Chapter 15. Policies: Controlling Precision, Error Handling etc">Policy</a> argument is optional and can | |
157 | be used to control the behaviour of the function: how it handles errors, | |
158 | what level of precision to use etc. Refer to the <a class="link" href="../../policy.html" title="Chapter 15. Policies: Controlling Precision, Error Handling etc">policy | |
159 | documentation for more details</a>. | |
160 | </p> | |
161 | <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> | |
162 | <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_inv</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">q</span><span class="special">);</span> | |
163 | ||
164 | <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 15. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">></span> | |
165 | <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_inv</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">q</span><span class="special">,</span> <span class="keyword">const</span> <a class="link" href="../../policy.html" title="Chapter 15. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&);</span> | |
166 | ||
167 | <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> <span class="identifier">T4</span><span class="special">></span> | |
168 | <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_inv</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">q</span><span class="special">,</span> <span class="identifier">T4</span><span class="special">*</span> <span class="identifier">py</span><span class="special">);</span> | |
169 | ||
170 | <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> <span class="identifier">T4</span><span class="special">,</span> <span class="keyword">class</span> <a class="link" href="../../policy.html" title="Chapter 15. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">></span> | |
171 | <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_inv</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">q</span><span class="special">,</span> <span class="identifier">T4</span><span class="special">*</span> <span class="identifier">py</span><span class="special">,</span> <span class="keyword">const</span> <a class="link" href="../../policy.html" title="Chapter 15. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&);</span> | |
172 | </pre> | |
173 | <p> | |
174 | Returns a value <span class="emphasis"><em>x</em></span> such that: <code class="computeroutput"><span class="identifier">q</span> | |
175 | <span class="special">=</span> <span class="identifier">ibetac</span><span class="special">(</span><span class="identifier">a</span><span class="special">,</span> | |
176 | <span class="identifier">b</span><span class="special">,</span> <span class="identifier">x</span><span class="special">);</span></code> and | |
177 | sets <code class="computeroutput"><span class="special">*</span><span class="identifier">py</span> | |
178 | <span class="special">=</span> <span class="number">1</span> <span class="special">-</span> <span class="identifier">x</span></code> when | |
179 | the <code class="computeroutput"><span class="identifier">py</span></code> parameter is provided | |
180 | and is non-null. Note that internally this function computes whichever is | |
181 | the smaller of <code class="computeroutput"><span class="identifier">x</span></code> and <code class="computeroutput"><span class="number">1</span><span class="special">-</span><span class="identifier">x</span></code>, | |
182 | and therefore the value assigned to <code class="computeroutput"><span class="special">*</span><span class="identifier">py</span></code> is free from cancellation errors. That | |
183 | means that even if the function returns <code class="computeroutput"><span class="number">1</span></code>, | |
184 | the value stored in <code class="computeroutput"><span class="special">*</span><span class="identifier">py</span></code> | |
185 | may be non-zero, albeit very small. | |
186 | </p> | |
187 | <p> | |
188 | Requires: <span class="emphasis"><em>a,b > 0</em></span> and <span class="emphasis"><em>0 <= q <= 1</em></span>. | |
189 | </p> | |
190 | <p> | |
191 | The final <a class="link" href="../../policy.html" title="Chapter 15. Policies: Controlling Precision, Error Handling etc">Policy</a> argument is optional and can | |
192 | be used to control the behaviour of the function: how it handles errors, | |
193 | what level of precision to use etc. Refer to the <a class="link" href="../../policy.html" title="Chapter 15. Policies: Controlling Precision, Error Handling etc">policy | |
194 | documentation for more details</a>. | |
195 | </p> | |
196 | <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> | |
197 | <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_inva</span><span class="special">(</span><span class="identifier">T1</span> <span class="identifier">b</span><span class="special">,</span> <span class="identifier">T2</span> <span class="identifier">x</span><span class="special">,</span> <span class="identifier">T3</span> <span class="identifier">p</span><span class="special">);</span> | |
198 | ||
199 | <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 15. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">></span> | |
200 | <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_inva</span><span class="special">(</span><span class="identifier">T1</span> <span class="identifier">b</span><span class="special">,</span> <span class="identifier">T2</span> <span class="identifier">x</span><span class="special">,</span> <span class="identifier">T3</span> <span class="identifier">p</span><span class="special">,</span> <span class="keyword">const</span> <a class="link" href="../../policy.html" title="Chapter 15. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&);</span> | |
201 | </pre> | |
202 | <p> | |
203 | Returns a value <span class="emphasis"><em>a</em></span> such that: <code class="computeroutput"><span class="identifier">p</span> | |
204 | <span class="special">=</span> <span class="identifier">ibeta</span><span class="special">(</span><span class="identifier">a</span><span class="special">,</span> | |
205 | <span class="identifier">b</span><span class="special">,</span> <span class="identifier">x</span><span class="special">);</span></code> | |
206 | </p> | |
207 | <p> | |
208 | Requires: <span class="emphasis"><em>b > 0</em></span>, <span class="emphasis"><em>0 < x < 1</em></span> | |
209 | and <span class="emphasis"><em>0 <= p <= 1</em></span>. | |
210 | </p> | |
211 | <p> | |
212 | The final <a class="link" href="../../policy.html" title="Chapter 15. Policies: Controlling Precision, Error Handling etc">Policy</a> argument is optional and can | |
213 | be used to control the behaviour of the function: how it handles errors, | |
214 | what level of precision to use etc. Refer to the <a class="link" href="../../policy.html" title="Chapter 15. Policies: Controlling Precision, Error Handling etc">policy | |
215 | documentation for more details</a>. | |
216 | </p> | |
217 | <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> | |
218 | <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_inva</span><span class="special">(</span><span class="identifier">T1</span> <span class="identifier">b</span><span class="special">,</span> <span class="identifier">T2</span> <span class="identifier">x</span><span class="special">,</span> <span class="identifier">T3</span> <span class="identifier">p</span><span class="special">);</span> | |
219 | ||
220 | <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 15. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">></span> | |
221 | <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_inva</span><span class="special">(</span><span class="identifier">T1</span> <span class="identifier">b</span><span class="special">,</span> <span class="identifier">T2</span> <span class="identifier">x</span><span class="special">,</span> <span class="identifier">T3</span> <span class="identifier">p</span><span class="special">,</span> <span class="keyword">const</span> <a class="link" href="../../policy.html" title="Chapter 15. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&);</span> | |
222 | </pre> | |
223 | <p> | |
224 | Returns a value <span class="emphasis"><em>a</em></span> such that: <code class="computeroutput"><span class="identifier">q</span> | |
225 | <span class="special">=</span> <span class="identifier">ibetac</span><span class="special">(</span><span class="identifier">a</span><span class="special">,</span> | |
226 | <span class="identifier">b</span><span class="special">,</span> <span class="identifier">x</span><span class="special">);</span></code> | |
227 | </p> | |
228 | <p> | |
229 | Requires: <span class="emphasis"><em>b > 0</em></span>, <span class="emphasis"><em>0 < x < 1</em></span> | |
230 | and <span class="emphasis"><em>0 <= q <= 1</em></span>. | |
231 | </p> | |
232 | <p> | |
233 | The final <a class="link" href="../../policy.html" title="Chapter 15. Policies: Controlling Precision, Error Handling etc">Policy</a> argument is optional and can | |
234 | be used to control the behaviour of the function: how it handles errors, | |
235 | what level of precision to use etc. Refer to the <a class="link" href="../../policy.html" title="Chapter 15. Policies: Controlling Precision, Error Handling etc">policy | |
236 | documentation for more details</a>. | |
237 | </p> | |
238 | <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> | |
239 | <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_invb</span><span class="special">(</span><span class="identifier">T1</span> <span class="identifier">b</span><span class="special">,</span> <span class="identifier">T2</span> <span class="identifier">x</span><span class="special">,</span> <span class="identifier">T3</span> <span class="identifier">p</span><span class="special">);</span> | |
240 | ||
241 | <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 15. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">></span> | |
242 | <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_invb</span><span class="special">(</span><span class="identifier">T1</span> <span class="identifier">b</span><span class="special">,</span> <span class="identifier">T2</span> <span class="identifier">x</span><span class="special">,</span> <span class="identifier">T3</span> <span class="identifier">p</span><span class="special">,</span> <span class="keyword">const</span> <a class="link" href="../../policy.html" title="Chapter 15. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&);</span> | |
243 | </pre> | |
244 | <p> | |
245 | Returns a value <span class="emphasis"><em>b</em></span> such that: <code class="computeroutput"><span class="identifier">p</span> | |
246 | <span class="special">=</span> <span class="identifier">ibeta</span><span class="special">(</span><span class="identifier">a</span><span class="special">,</span> | |
247 | <span class="identifier">b</span><span class="special">,</span> <span class="identifier">x</span><span class="special">);</span></code> | |
248 | </p> | |
249 | <p> | |
250 | Requires: <span class="emphasis"><em>a > 0</em></span>, <span class="emphasis"><em>0 < x < 1</em></span> | |
251 | and <span class="emphasis"><em>0 <= p <= 1</em></span>. | |
252 | </p> | |
253 | <p> | |
254 | The final <a class="link" href="../../policy.html" title="Chapter 15. Policies: Controlling Precision, Error Handling etc">Policy</a> argument is optional and can | |
255 | be used to control the behaviour of the function: how it handles errors, | |
256 | what level of precision to use etc. Refer to the <a class="link" href="../../policy.html" title="Chapter 15. Policies: Controlling Precision, Error Handling etc">policy | |
257 | documentation for more details</a>. | |
258 | </p> | |
259 | <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> | |
260 | <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_invb</span><span class="special">(</span><span class="identifier">T1</span> <span class="identifier">b</span><span class="special">,</span> <span class="identifier">T2</span> <span class="identifier">x</span><span class="special">,</span> <span class="identifier">T3</span> <span class="identifier">p</span><span class="special">);</span> | |
261 | ||
262 | <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 15. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">></span> | |
263 | <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_invb</span><span class="special">(</span><span class="identifier">T1</span> <span class="identifier">b</span><span class="special">,</span> <span class="identifier">T2</span> <span class="identifier">x</span><span class="special">,</span> <span class="identifier">T3</span> <span class="identifier">p</span><span class="special">,</span> <span class="keyword">const</span> <a class="link" href="../../policy.html" title="Chapter 15. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&);</span> | |
264 | </pre> | |
265 | <p> | |
266 | Returns a value <span class="emphasis"><em>b</em></span> such that: <code class="computeroutput"><span class="identifier">q</span> | |
267 | <span class="special">=</span> <span class="identifier">ibetac</span><span class="special">(</span><span class="identifier">a</span><span class="special">,</span> | |
268 | <span class="identifier">b</span><span class="special">,</span> <span class="identifier">x</span><span class="special">);</span></code> | |
269 | </p> | |
270 | <p> | |
271 | Requires: <span class="emphasis"><em>a > 0</em></span>, <span class="emphasis"><em>0 < x < 1</em></span> | |
272 | and <span class="emphasis"><em>0 <= q <= 1</em></span>. | |
273 | </p> | |
274 | <p> | |
275 | The final <a class="link" href="../../policy.html" title="Chapter 15. Policies: Controlling Precision, Error Handling etc">Policy</a> argument is optional and can | |
276 | be used to control the behaviour of the function: how it handles errors, | |
277 | what level of precision to use etc. Refer to the <a class="link" href="../../policy.html" title="Chapter 15. Policies: Controlling Precision, Error Handling etc">policy | |
278 | documentation for more details</a>. | |
279 | </p> | |
280 | <h5> | |
281 | <a name="math_toolkit.sf_beta.ibeta_inv_function.h1"></a> | |
282 | <span class="phrase"><a name="math_toolkit.sf_beta.ibeta_inv_function.accuracy"></a></span><a class="link" href="ibeta_inv_function.html#math_toolkit.sf_beta.ibeta_inv_function.accuracy">Accuracy</a> | |
283 | </h5> | |
284 | <p> | |
285 | The accuracy of these functions should closely follow that of the regular | |
286 | forward incomplete beta functions. However, note that in some parts of their | |
287 | domain, these functions can be extremely sensitive to changes in input, particularly | |
288 | when the argument <span class="emphasis"><em>p</em></span> (or it's complement <span class="emphasis"><em>q</em></span>) | |
289 | is very close to <code class="computeroutput"><span class="number">0</span></code> or <code class="computeroutput"><span class="number">1</span></code>. | |
290 | </p> | |
291 | <p> | |
292 | Comparisons to other libraries are shown below, note that our test data exercises | |
293 | some rather extreme cases in the incomplete beta function which many other | |
294 | libraries fail to handle: | |
295 | </p> | |
296 | <div class="table"> | |
297 | <a name="math_toolkit.sf_beta.ibeta_inv_function.table_ibeta_inv"></a><p class="title"><b>Table 6.22. Error rates for ibeta_inv</b></p> | |
298 | <div class="table-contents"><table class="table" summary="Error rates for ibeta_inv"> | |
299 | <colgroup> | |
300 | <col> | |
301 | <col> | |
302 | <col> | |
303 | <col> | |
304 | <col> | |
305 | </colgroup> | |
306 | <thead><tr> | |
307 | <th> | |
308 | </th> | |
309 | <th> | |
310 | <p> | |
311 | Microsoft Visual C++ version 12.0<br> Win32<br> double | |
312 | </p> | |
313 | </th> | |
314 | <th> | |
315 | <p> | |
316 | GNU C++ version 5.1.0<br> linux<br> double | |
317 | </p> | |
318 | </th> | |
319 | <th> | |
320 | <p> | |
321 | GNU C++ version 5.1.0<br> linux<br> long double | |
322 | </p> | |
323 | </th> | |
324 | <th> | |
325 | <p> | |
326 | Sun compiler version 0x5130<br> Sun Solaris<br> long double | |
327 | </p> | |
328 | </th> | |
329 | </tr></thead> | |
330 | <tbody><tr> | |
331 | <td> | |
332 | <p> | |
333 | Inverse incomplete beta | |
334 | </p> | |
335 | </td> | |
336 | <td> | |
337 | <p> | |
338 | <span class="blue">Max = 7.08e+003ε (Mean = 244ε)</span> | |
339 | </p> | |
340 | </td> | |
341 | <td> | |
342 | <p> | |
343 | <span class="blue">Max = 3.21ε (Mean = 0.158ε)</span><br> <br> | |
344 | (<span class="emphasis"><em>Rmath 3.0.2:</em></span> <span class="red">Max = +INFε (Mean | |
345 | = +INFε) <a class="link" href="../logs_and_tables/logs.html#errors_GNU_C_version_5_1_0_linux_double_ibeta_inv_Rmath_3_0_2_Inverse_incomplete_beta">And | |
346 | other failures.</a>)</span><br> (<span class="emphasis"><em>Cephes:</em></span> | |
347 | <span class="red">Max = 1.28e+10ε (Mean = 5.17e+08ε))</span> | |
348 | </p> | |
349 | </td> | |
350 | <td> | |
351 | <p> | |
352 | <span class="blue">Max = 4.53e+04ε (Mean = 2.93e+03ε)</span> | |
353 | </p> | |
354 | </td> | |
355 | <td> | |
356 | <p> | |
357 | <span class="blue">Max = 4.22e+04ε (Mean = 2.8e+03ε)</span> | |
358 | </p> | |
359 | </td> | |
360 | </tr></tbody> | |
361 | </table></div> | |
362 | </div> | |
363 | <br class="table-break"><div class="table"> | |
364 | <a name="math_toolkit.sf_beta.ibeta_inv_function.table_ibetac_inv"></a><p class="title"><b>Table 6.23. Error rates for ibetac_inv</b></p> | |
365 | <div class="table-contents"><table class="table" summary="Error rates for ibetac_inv"> | |
366 | <colgroup> | |
367 | <col> | |
368 | <col> | |
369 | <col> | |
370 | <col> | |
371 | <col> | |
372 | </colgroup> | |
373 | <thead><tr> | |
374 | <th> | |
375 | </th> | |
376 | <th> | |
377 | <p> | |
378 | Microsoft Visual C++ version 12.0<br> Win32<br> double | |
379 | </p> | |
380 | </th> | |
381 | <th> | |
382 | <p> | |
383 | GNU C++ version 5.1.0<br> linux<br> double | |
384 | </p> | |
385 | </th> | |
386 | <th> | |
387 | <p> | |
388 | GNU C++ version 5.1.0<br> linux<br> long double | |
389 | </p> | |
390 | </th> | |
391 | <th> | |
392 | <p> | |
393 | Sun compiler version 0x5130<br> Sun Solaris<br> long double | |
394 | </p> | |
395 | </th> | |
396 | </tr></thead> | |
397 | <tbody><tr> | |
398 | <td> | |
399 | <p> | |
400 | Inverse incomplete beta | |
401 | </p> | |
402 | </td> | |
403 | <td> | |
404 | <p> | |
405 | <span class="blue">Max = 5.53e+003ε (Mean = 220ε)</span> | |
406 | </p> | |
407 | </td> | |
408 | <td> | |
409 | <p> | |
410 | <span class="blue">Max = 1.71ε (Mean = 0.108ε)</span><br> <br> | |
411 | (<span class="emphasis"><em>Rmath 3.0.2:</em></span> <span class="red">Max = +INFε (Mean | |
412 | = +INFε) <a class="link" href="../logs_and_tables/logs.html#errors_GNU_C_version_5_1_0_linux_double_ibetac_inv_Rmath_3_0_2_Inverse_incomplete_beta">And | |
413 | other failures.</a>)</span><br> (<span class="emphasis"><em>Cephes:</em></span> | |
414 | Max = 1.71ε (Mean = 0.108ε)) | |
415 | </p> | |
416 | </td> | |
417 | <td> | |
418 | <p> | |
419 | <span class="blue">Max = 6.17e+04ε (Mean = 3.77e+03ε)</span> | |
420 | </p> | |
421 | </td> | |
422 | <td> | |
423 | <p> | |
424 | <span class="blue">Max = 5.15e+04ε (Mean = 3.51e+03ε)</span> | |
425 | </p> | |
426 | </td> | |
427 | </tr></tbody> | |
428 | </table></div> | |
429 | </div> | |
430 | <br class="table-break"><div class="table"> | |
431 | <a name="math_toolkit.sf_beta.ibeta_inv_function.table_ibeta_inva"></a><p class="title"><b>Table 6.24. Error rates for ibeta_inva</b></p> | |
432 | <div class="table-contents"><table class="table" summary="Error rates for ibeta_inva"> | |
433 | <colgroup> | |
434 | <col> | |
435 | <col> | |
436 | <col> | |
437 | <col> | |
438 | <col> | |
439 | </colgroup> | |
440 | <thead><tr> | |
441 | <th> | |
442 | </th> | |
443 | <th> | |
444 | <p> | |
445 | Microsoft Visual C++ version 12.0<br> Win32<br> double | |
446 | </p> | |
447 | </th> | |
448 | <th> | |
449 | <p> | |
450 | GNU C++ version 5.1.0<br> linux<br> double | |
451 | </p> | |
452 | </th> | |
453 | <th> | |
454 | <p> | |
455 | GNU C++ version 5.1.0<br> linux<br> long double | |
456 | </p> | |
457 | </th> | |
458 | <th> | |
459 | <p> | |
460 | Sun compiler version 0x5130<br> Sun Solaris<br> long double | |
461 | </p> | |
462 | </th> | |
463 | </tr></thead> | |
464 | <tbody><tr> | |
465 | <td> | |
466 | <p> | |
467 | Inverse incomplete beta | |
468 | </p> | |
469 | </td> | |
470 | <td> | |
471 | <p> | |
472 | <span class="blue">Max = 395ε (Mean = 24.7ε)</span> | |
473 | </p> | |
474 | </td> | |
475 | <td> | |
476 | <p> | |
477 | <span class="blue">Max = 0.602ε (Mean = 0.0239ε)</span> | |
478 | </p> | |
479 | </td> | |
480 | <td> | |
481 | <p> | |
482 | <span class="blue">Max = 377ε (Mean = 25.1ε)</span> | |
483 | </p> | |
484 | </td> | |
485 | <td> | |
486 | <p> | |
487 | <span class="blue">Max = 438ε (Mean = 31.6ε)</span> | |
488 | </p> | |
489 | </td> | |
490 | </tr></tbody> | |
491 | </table></div> | |
492 | </div> | |
493 | <br class="table-break"><div class="table"> | |
494 | <a name="math_toolkit.sf_beta.ibeta_inv_function.table_ibetac_inva"></a><p class="title"><b>Table 6.25. Error rates for ibetac_inva</b></p> | |
495 | <div class="table-contents"><table class="table" summary="Error rates for ibetac_inva"> | |
496 | <colgroup> | |
497 | <col> | |
498 | <col> | |
499 | <col> | |
500 | <col> | |
501 | <col> | |
502 | </colgroup> | |
503 | <thead><tr> | |
504 | <th> | |
505 | </th> | |
506 | <th> | |
507 | <p> | |
508 | Microsoft Visual C++ version 12.0<br> Win32<br> double | |
509 | </p> | |
510 | </th> | |
511 | <th> | |
512 | <p> | |
513 | GNU C++ version 5.1.0<br> linux<br> double | |
514 | </p> | |
515 | </th> | |
516 | <th> | |
517 | <p> | |
518 | GNU C++ version 5.1.0<br> linux<br> long double | |
519 | </p> | |
520 | </th> | |
521 | <th> | |
522 | <p> | |
523 | Sun compiler version 0x5130<br> Sun Solaris<br> long double | |
524 | </p> | |
525 | </th> | |
526 | </tr></thead> | |
527 | <tbody><tr> | |
528 | <td> | |
529 | <p> | |
530 | Inverse incomplete beta | |
531 | </p> | |
532 | </td> | |
533 | <td> | |
534 | <p> | |
535 | <span class="blue">Max = 408ε (Mean = 27.8ε)</span> | |
536 | </p> | |
537 | </td> | |
538 | <td> | |
539 | <p> | |
540 | <span class="blue">Max = 0.683ε (Mean = 0.0271ε)</span> | |
541 | </p> | |
542 | </td> | |
543 | <td> | |
544 | <p> | |
545 | <span class="blue">Max = 382ε (Mean = 22.2ε)</span> | |
546 | </p> | |
547 | </td> | |
548 | <td> | |
549 | <p> | |
550 | <span class="blue">Max = 315ε (Mean = 23.7ε)</span> | |
551 | </p> | |
552 | </td> | |
553 | </tr></tbody> | |
554 | </table></div> | |
555 | </div> | |
556 | <br class="table-break"><div class="table"> | |
557 | <a name="math_toolkit.sf_beta.ibeta_inv_function.table_ibeta_invb"></a><p class="title"><b>Table 6.26. Error rates for ibeta_invb</b></p> | |
558 | <div class="table-contents"><table class="table" summary="Error rates for ibeta_invb"> | |
559 | <colgroup> | |
560 | <col> | |
561 | <col> | |
562 | <col> | |
563 | <col> | |
564 | <col> | |
565 | </colgroup> | |
566 | <thead><tr> | |
567 | <th> | |
568 | </th> | |
569 | <th> | |
570 | <p> | |
571 | Microsoft Visual C++ version 12.0<br> Win32<br> double | |
572 | </p> | |
573 | </th> | |
574 | <th> | |
575 | <p> | |
576 | GNU C++ version 5.1.0<br> linux<br> double | |
577 | </p> | |
578 | </th> | |
579 | <th> | |
580 | <p> | |
581 | GNU C++ version 5.1.0<br> linux<br> long double | |
582 | </p> | |
583 | </th> | |
584 | <th> | |
585 | <p> | |
586 | Sun compiler version 0x5130<br> Sun Solaris<br> long double | |
587 | </p> | |
588 | </th> | |
589 | </tr></thead> | |
590 | <tbody><tr> | |
591 | <td> | |
592 | <p> | |
593 | Inverse incomplete beta | |
594 | </p> | |
595 | </td> | |
596 | <td> | |
597 | <p> | |
598 | <span class="blue">Max = 409ε (Mean = 17.9ε)</span> | |
599 | </p> | |
600 | </td> | |
601 | <td> | |
602 | <p> | |
603 | <span class="blue">Max = 0.836ε (Mean = 0.0491ε)</span> | |
604 | </p> | |
605 | </td> | |
606 | <td> | |
607 | <p> | |
608 | <span class="blue">Max = 407ε (Mean = 27.2ε)</span> | |
609 | </p> | |
610 | </td> | |
611 | <td> | |
612 | <p> | |
613 | <span class="blue">Max = 407ε (Mean = 24.4ε)</span> | |
614 | </p> | |
615 | </td> | |
616 | </tr></tbody> | |
617 | </table></div> | |
618 | </div> | |
619 | <br class="table-break"><div class="table"> | |
620 | <a name="math_toolkit.sf_beta.ibeta_inv_function.table_ibetac_invb"></a><p class="title"><b>Table 6.27. Error rates for ibetac_invb</b></p> | |
621 | <div class="table-contents"><table class="table" summary="Error rates for ibetac_invb"> | |
622 | <colgroup> | |
623 | <col> | |
624 | <col> | |
625 | <col> | |
626 | <col> | |
627 | <col> | |
628 | </colgroup> | |
629 | <thead><tr> | |
630 | <th> | |
631 | </th> | |
632 | <th> | |
633 | <p> | |
634 | Microsoft Visual C++ version 12.0<br> Win32<br> double | |
635 | </p> | |
636 | </th> | |
637 | <th> | |
638 | <p> | |
639 | GNU C++ version 5.1.0<br> linux<br> double | |
640 | </p> | |
641 | </th> | |
642 | <th> | |
643 | <p> | |
644 | GNU C++ version 5.1.0<br> linux<br> long double | |
645 | </p> | |
646 | </th> | |
647 | <th> | |
648 | <p> | |
649 | Sun compiler version 0x5130<br> Sun Solaris<br> long double | |
650 | </p> | |
651 | </th> | |
652 | </tr></thead> | |
653 | <tbody><tr> | |
654 | <td> | |
655 | <p> | |
656 | Inverse incomplete beta | |
657 | </p> | |
658 | </td> | |
659 | <td> | |
660 | <p> | |
661 | <span class="blue">Max = 329ε (Mean = 18.2ε)</span> | |
662 | </p> | |
663 | </td> | |
664 | <td> | |
665 | <p> | |
666 | <span class="blue">Max = 0.724ε (Mean = 0.0303ε)</span> | |
667 | </p> | |
668 | </td> | |
669 | <td> | |
670 | <p> | |
671 | <span class="blue">Max = 317ε (Mean = 19.7ε)</span> | |
672 | </p> | |
673 | </td> | |
674 | <td> | |
675 | <p> | |
676 | <span class="blue">Max = 369ε (Mean = 21.7ε)</span> | |
677 | </p> | |
678 | </td> | |
679 | </tr></tbody> | |
680 | </table></div> | |
681 | </div> | |
682 | <br class="table-break"><h5> | |
683 | <a name="math_toolkit.sf_beta.ibeta_inv_function.h2"></a> | |
684 | <span class="phrase"><a name="math_toolkit.sf_beta.ibeta_inv_function.testing"></a></span><a class="link" href="ibeta_inv_function.html#math_toolkit.sf_beta.ibeta_inv_function.testing">Testing</a> | |
685 | </h5> | |
686 | <p> | |
687 | There are two sets of tests: | |
688 | </p> | |
689 | <div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: disc; "> | |
690 | <li class="listitem"> | |
691 | Basic sanity checks attempt to "round-trip" from <span class="emphasis"><em>a, | |
692 | b</em></span> and <span class="emphasis"><em>x</em></span> to <span class="emphasis"><em>p</em></span> or | |
693 | <span class="emphasis"><em>q</em></span> and back again. These tests have quite generous | |
694 | tolerances: in general both the incomplete beta and its inverses change | |
695 | so rapidly, that round tripping to more than a couple of significant | |
696 | digits isn't possible. This is especially true when <span class="emphasis"><em>p</em></span> | |
697 | or <span class="emphasis"><em>q</em></span> is very near one: in this case there isn't | |
698 | enough "information content" in the input to the inverse function | |
699 | to get back where you started. | |
700 | </li> | |
701 | <li class="listitem"> | |
702 | Accuracy checks using high precision test values. These measure the accuracy | |
703 | of the result, given exact input values. | |
704 | </li> | |
705 | </ul></div> | |
706 | <h5> | |
707 | <a name="math_toolkit.sf_beta.ibeta_inv_function.h3"></a> | |
708 | <span class="phrase"><a name="math_toolkit.sf_beta.ibeta_inv_function.implementation_of_ibeta_inv_and_"></a></span><a class="link" href="ibeta_inv_function.html#math_toolkit.sf_beta.ibeta_inv_function.implementation_of_ibeta_inv_and_">Implementation | |
709 | of ibeta_inv and ibetac_inv</a> | |
710 | </h5> | |
711 | <p> | |
712 | These two functions share a common implementation. | |
713 | </p> | |
714 | <p> | |
715 | First an initial approximation to x is computed then the last few bits are | |
716 | cleaned up using <a href="http://en.wikipedia.org/wiki/Simple_rational_approximation" target="_top">Halley | |
717 | iteration</a>. The iteration limit is set to 1/2 of the number of bits | |
718 | in T, which by experiment is sufficient to ensure that the inverses are at | |
719 | least as accurate as the normal incomplete beta functions. Up to 5 iterations | |
720 | may be required in extreme cases, although normally only one or two are required. | |
721 | Further, the number of iterations required decreases with increasing <span class="emphasis"><em>a</em></span> | |
722 | and <span class="emphasis"><em>b</em></span> (which generally form the more important use cases). | |
723 | </p> | |
724 | <p> | |
725 | The initial guesses used for iteration are obtained as follows: | |
726 | </p> | |
727 | <p> | |
728 | Firstly recall that: | |
729 | </p> | |
730 | <p> | |
731 | <span class="inlinemediaobject"><img src="../../../equations/ibeta_inv5.svg"></span> | |
732 | </p> | |
733 | <p> | |
734 | We may wish to start from either p or q, and to calculate either x or y. | |
735 | In addition at any stage we can exchange a for b, p for q, and x for y if | |
736 | it results in a more manageable problem. | |
737 | </p> | |
738 | <p> | |
739 | For <code class="computeroutput"><span class="identifier">a</span><span class="special">+</span><span class="identifier">b</span> <span class="special">>=</span> <span class="number">5</span></code> the initial guess is computed using the methods | |
740 | described in: | |
741 | </p> | |
742 | <p> | |
743 | Asymptotic Inversion of the Incomplete Beta Function, by N. M. <a href="http://homepages.cwi.nl/~nicot/" target="_top">Temme</a>. | |
744 | Journal of Computational and Applied Mathematics 41 (1992) 145-157. | |
745 | </p> | |
746 | <p> | |
747 | The nearly symmetrical case (section 2 of the paper) is used for | |
748 | </p> | |
749 | <p> | |
750 | <span class="inlinemediaobject"><img src="../../../equations/ibeta_inv2.svg"></span> | |
751 | </p> | |
752 | <p> | |
753 | and involves solving the inverse error function first. The method is accurate | |
754 | to at least 2 decimal digits when <code class="literal">a = 5</code> rising to at least | |
755 | 8 digits when <code class="literal">a = 10<sup>5</sup></code>. | |
756 | </p> | |
757 | <p> | |
758 | The general error function case (section 3 of the paper) is used for | |
759 | </p> | |
760 | <p> | |
761 | <span class="inlinemediaobject"><img src="../../../equations/ibeta_inv3.svg"></span> | |
762 | </p> | |
763 | <p> | |
764 | and again expresses the inverse incomplete beta in terms of the inverse of | |
765 | the error function. The method is accurate to at least 2 decimal digits when | |
766 | <code class="literal">a+b = 5</code> rising to 11 digits when <code class="literal">a+b = 10<sup>5</sup></code>. | |
767 | However, when the result is expected to be very small, and when a+b is also | |
768 | small, then its accuracy tails off, in this case when p<sup>1/a</sup> < 0.0025 then | |
769 | it is better to use the following as an initial estimate: | |
770 | </p> | |
771 | <p> | |
772 | <span class="inlinemediaobject"><img src="../../../equations/ibeta_inv4.svg"></span> | |
773 | </p> | |
774 | <p> | |
775 | Finally the for all other cases where <code class="computeroutput"><span class="identifier">a</span><span class="special">+</span><span class="identifier">b</span> <span class="special">></span> | |
776 | <span class="number">5</span></code> the method of section 4 of the paper | |
777 | is used. This expresses the inverse incomplete beta in terms of the inverse | |
778 | of the incomplete gamma function, and is therefore significantly more expensive | |
779 | to compute than the other cases. However the method is accurate to at least | |
780 | 3 decimal digits when <code class="literal">a = 5</code> rising to at least 10 digits | |
781 | when <code class="literal">a = 10<sup>5</sup></code>. This method is limited to a > b, and | |
782 | therefore we need to perform an exchange a for b, p for q and x for y when | |
783 | this is not the case. In addition when p is close to 1 the method is inaccurate | |
784 | should we actually want y rather than x as output. Therefore when q is small | |
785 | (<code class="literal">q<sup>1/p</sup> < 10<sup>-3</sup></code>) we use: | |
786 | </p> | |
787 | <p> | |
788 | <span class="inlinemediaobject"><img src="../../../equations/ibeta_inv6.svg"></span> | |
789 | </p> | |
790 | <p> | |
791 | which is both cheaper to compute than the full method, and a more accurate | |
792 | estimate on q. | |
793 | </p> | |
794 | <p> | |
795 | When a and b are both small there is a distinct lack of information in the | |
796 | literature on how to proceed. I am extremely grateful to Prof Nico Temme | |
797 | who provided the following information with a great deal of patience and | |
798 | explanation on his part. Any errors that follow are entirely my own, and | |
799 | not Prof Temme's. | |
800 | </p> | |
801 | <p> | |
802 | When a and b are both less than 1, then there is a point of inflection in | |
803 | the incomplete beta at point <code class="computeroutput"><span class="identifier">xs</span> | |
804 | <span class="special">=</span> <span class="special">(</span><span class="number">1</span> <span class="special">-</span> <span class="identifier">a</span><span class="special">)</span> <span class="special">/</span> <span class="special">(</span><span class="number">2</span> <span class="special">-</span> <span class="identifier">a</span> | |
805 | <span class="special">-</span> <span class="identifier">b</span><span class="special">)</span></code>. Therefore if <code class="literal">p > I<sub>x</sub>(a,b)</code> | |
806 | we swap a for b, p for q and x for y, so that now we always look for a point | |
807 | x below the point of inflection <code class="computeroutput"><span class="identifier">xs</span></code>, | |
808 | and on a convex curve. An initial estimate for x is made with: | |
809 | </p> | |
810 | <p> | |
811 | <span class="inlinemediaobject"><img src="../../../equations/ibeta_inv7.svg"></span> | |
812 | </p> | |
813 | <p> | |
814 | which is provably below the true value for x: <a href="http://en.wikipedia.org/wiki/Newton%27s_method" target="_top">Newton | |
815 | iteration</a> will therefore smoothly converge on x without problems | |
816 | caused by overshooting etc. | |
817 | </p> | |
818 | <p> | |
819 | When a and b are both greater than 1, but a+b is too small to use the other | |
820 | methods mentioned above, we proceed as follows. Observe that there is a point | |
821 | of inflection in the incomplete beta at <code class="computeroutput"><span class="identifier">xs</span> | |
822 | <span class="special">=</span> <span class="special">(</span><span class="number">1</span> <span class="special">-</span> <span class="identifier">a</span><span class="special">)</span> <span class="special">/</span> <span class="special">(</span><span class="number">2</span> <span class="special">-</span> <span class="identifier">a</span> | |
823 | <span class="special">-</span> <span class="identifier">b</span><span class="special">)</span></code>. Therefore if <code class="literal">p > I<sub>x</sub>(a,b)</code> | |
824 | we swap a for b, p for q and x for y, so that now we always look for a point | |
825 | x below the point of inflection <code class="computeroutput"><span class="identifier">xs</span></code>, | |
826 | and on a concave curve. An initial estimate for x is made with: | |
827 | </p> | |
828 | <p> | |
829 | <span class="inlinemediaobject"><img src="../../../equations/ibeta_inv4.svg"></span> | |
830 | </p> | |
831 | <p> | |
832 | which can be improved somewhat to: | |
833 | </p> | |
834 | <p> | |
835 | <span class="inlinemediaobject"><img src="../../../equations/ibeta_inv1.svg"></span> | |
836 | </p> | |
837 | <p> | |
838 | when b and x are both small (I've used b < a and x < 0.2). This actually | |
839 | under-estimates x, which drops us on the wrong side of x for Newton iteration | |
840 | to converge monotonically. However, use of higher derivatives and Halley | |
841 | iteration keeps everything under control. | |
842 | </p> | |
843 | <p> | |
844 | The final case to be considered if when one of a and b is less than or equal | |
845 | to 1, and the other greater that 1. Here, if b < a we swap a for b, p | |
846 | for q and x for y. Now the curve of the incomplete beta is convex with no | |
847 | points of inflection in [0,1]. For small p, x can be estimated using | |
848 | </p> | |
849 | <p> | |
850 | <span class="inlinemediaobject"><img src="../../../equations/ibeta_inv4.svg"></span> | |
851 | </p> | |
852 | <p> | |
853 | which under-estimates x, and drops us on the right side of the true value | |
854 | for Newton iteration to converge monotonically. However, when p is large | |
855 | this can quite badly underestimate x. This is especially an issue when we | |
856 | really want to find y, in which case this method can be an arbitrary number | |
857 | of order of magnitudes out, leading to very poor convergence during iteration. | |
858 | </p> | |
859 | <p> | |
860 | Things can be improved by considering the incomplete beta as a distorted | |
861 | quarter circle, and estimating y from: | |
862 | </p> | |
863 | <p> | |
864 | <span class="inlinemediaobject"><img src="../../../equations/ibeta_inv8.svg"></span> | |
865 | </p> | |
866 | <p> | |
867 | This doesn't guarantee that we will drop in on the right side of x for monotonic | |
868 | convergence, but it does get us close enough that Halley iteration rapidly | |
869 | converges on the true value. | |
870 | </p> | |
871 | <h5> | |
872 | <a name="math_toolkit.sf_beta.ibeta_inv_function.h4"></a> | |
873 | <span class="phrase"><a name="math_toolkit.sf_beta.ibeta_inv_function.implementation_of_inverses_on_th"></a></span><a class="link" href="ibeta_inv_function.html#math_toolkit.sf_beta.ibeta_inv_function.implementation_of_inverses_on_th">Implementation | |
874 | of inverses on the a and b parameters</a> | |
875 | </h5> | |
876 | <p> | |
877 | These four functions share a common implementation. | |
878 | </p> | |
879 | <p> | |
880 | First an initial approximation is computed for <span class="emphasis"><em>a</em></span> or | |
881 | <span class="emphasis"><em>b</em></span>: where possible this uses a Cornish-Fisher expansion | |
882 | for the negative binomial distribution to get within around 1 of the result. | |
883 | However, when <span class="emphasis"><em>a</em></span> or <span class="emphasis"><em>b</em></span> are very small | |
884 | the Cornish Fisher expansion is not usable, in this case the initial approximation | |
885 | is chosen so that I<sub>x</sub>(a, b) is near the middle of the range [0,1]. | |
886 | </p> | |
887 | <p> | |
888 | This initial guess is then used as a starting value for a generic root finding | |
889 | algorithm. The algorithm converges rapidly on the root once it has been bracketed, | |
890 | but bracketing the root may take several iterations. A better initial approximation | |
891 | for <span class="emphasis"><em>a</em></span> or <span class="emphasis"><em>b</em></span> would improve these | |
892 | functions quite substantially: currently 10-20 incomplete beta function invocations | |
893 | are required to find the root. | |
894 | </p> | |
895 | </div> | |
896 | <table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr> | |
897 | <td align="left"></td> | |
898 | <td align="right"><div class="copyright-footer">Copyright © 2006-2010, 2012-2014 Nikhar Agrawal, | |
899 | Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos, Hubert | |
900 | Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Johan Råde, Gautam Sewani, | |
901 | Benjamin Sobotta, Thijs van den Berg, Daryle Walker and Xiaogang Zhang<p> | |
902 | Distributed under the Boost Software License, Version 1.0. (See accompanying | |
903 | 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>) | |
904 | </p> | |
905 | </div></td> | |
906 | </tr></table> | |
907 | <hr> | |
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