<html> <head> <meta http-equiv="Content-Type" content="text/html; charset=US-ASCII"> <title>The Incomplete Beta Function Inverses</title> <link rel="stylesheet" href="../../../../../../../../doc/src/boostbook.css" type="text/css"> <meta name="generator" content="DocBook XSL Stylesheets V1.74.0"> <link rel="home" href="../../../index.html" title="Math Toolkit"> <link rel="up" href="../sf_beta.html" title="Beta Functions"> <link rel="prev" href="ibeta_function.html" title="Incomplete Beta Functions"> <link rel="next" href="beta_derivative.html" title="Derivative of the Incomplete Beta Function"> </head> <body bgcolor="white" text="black" link="#0000FF" vlink="#840084" alink="#0000FF"> <table cellpadding="2" width="100%"><tr> <td valign="top"><img alt="Boost C++ Libraries" width="277" height="86" src="../../../../../../../../boost.png"></td> <td align="center"><a href="../../../../../../../../index.html">Home</a></td> <td align="center"><a href="../../../../../../../../libs/libraries.htm">Libraries</a></td> <td align="center"><a href="http://www.boost.org/users/people.html">People</a></td> <td align="center"><a href="http://www.boost.org/users/faq.html">FAQ</a></td> <td align="center"><a href="../../../../../../../../more/index.htm">More</a></td> </tr></table> <hr> <div class="spirit-nav"> <a accesskey="p" href="ibeta_function.html"><img src="../../../../../../../../doc/src/images/prev.png" alt="Prev"></a><a accesskey="u" href="../sf_beta.html"><img src="../../../../../../../../doc/src/images/up.png" alt="Up"></a><a accesskey="h" href="../../../index.html"><img src="../../../../../../../../doc/src/images/home.png" alt="Home"></a><a accesskey="n" href="beta_derivative.html"><img src="../../../../../../../../doc/src/images/next.png" alt="Next"></a> </div> <div class="section" lang="en"> <div class="titlepage"><div><div><h4 class="title"> <a name="math_toolkit.special.sf_beta.ibeta_inv_function"></a><a class="link" href="ibeta_inv_function.html" title="The Incomplete Beta Function Inverses"> The Incomplete Beta Function Inverses</a> </h4></div></div></div> <p> </p> <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> </pre> <p> </p> <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> <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> <a class="link" href="../../main_overview/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">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="Policies">Policy</a><span class="special">></span> <a class="link" href="../../main_overview/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="Policies">Policy</a><span class="special">&);</span> <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> <a class="link" href="../../main_overview/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">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="Policies">Policy</a><span class="special">></span> <a class="link" href="../../main_overview/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="Policies">Policy</a><span class="special">&);</span> <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> <a class="link" href="../../main_overview/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">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="Policies">Policy</a><span class="special">></span> <a class="link" href="../../main_overview/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="Policies">Policy</a><span class="special">&);</span> <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> <a class="link" href="../../main_overview/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">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="Policies">Policy</a><span class="special">></span> <a class="link" href="../../main_overview/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="Policies">Policy</a><span class="special">&);</span> <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> <a class="link" href="../../main_overview/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">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="Policies">Policy</a><span class="special">></span> <a class="link" href="../../main_overview/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="Policies">Policy</a><span class="special">&);</span> <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> <a class="link" href="../../main_overview/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">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="Policies">Policy</a><span class="special">></span> <a class="link" href="../../main_overview/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="Policies">Policy</a><span class="special">&);</span> <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> <a class="link" href="../../main_overview/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">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="Policies">Policy</a><span class="special">></span> <a class="link" href="../../main_overview/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="Policies">Policy</a><span class="special">&);</span> <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> <a class="link" href="../../main_overview/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">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="Policies">Policy</a><span class="special">></span> <a class="link" href="../../main_overview/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="Policies">Policy</a><span class="special">&);</span> <span class="special">}}</span> <span class="comment">// namespaces </span></pre> <a name="math_toolkit.special.sf_beta.ibeta_inv_function.description"></a><h5> <a name="id1101233"></a> <a class="link" href="ibeta_inv_function.html#math_toolkit.special.sf_beta.ibeta_inv_function.description">Description</a> </h5> <p> There are six <a href="http://functions.wolfram.com/GammaBetaErf/" target="_top">incomplete beta function inverses</a> which allow you solve for any of the three parameters to the incomplete beta, starting from either the result of the incomplete beta (p) or its complement (q). </p> <p> </p> <p> The final <a class="link" href="../../policy.html" title="Policies">Policy</a> argument is optional and can be used to control the behaviour of the function: how it handles errors, what level of precision to use etc. Refer to the <a class="link" href="../../policy.html" title="Policies">policy documentation for more details</a>. </p> <p> </p> <div class="tip"><table border="0" summary="Tip"> <tr> <td rowspan="2" align="center" valign="top" width="25"><img alt="[Tip]" src="../../../../../../../../doc/src/images/tip.png"></td> <th align="left">Tip</th> </tr> <tr><td align="left" valign="top"> <p> When people normally talk about the inverse of the incomplete beta function, they are talking about inverting on parameter <span class="emphasis"><em>x</em></span>. These are implemented here as ibeta_inv and ibeta_inv, and are by far the most efficient of the inverses presented here. </p> <p> The inverses on the <span class="emphasis"><em>a</em></span> and <span class="emphasis"><em>b</em></span> parameters find use in some statistical applications, but have to be computed by rather brute force numerical techniques and are consequently several times slower. These are implemented here as ibeta_inva and ibeta_invb, and complement versions ibetac_inva and ibetac_invb. </p> </td></tr> </table></div> <p> The return type of these functions is computed using the <a class="link" href="../../main_overview/result_type.html" title="Calculation of the Type of the Result"><span class="emphasis"><em>result type calculation rules</em></span></a> when called with arguments T1...TN of different types. </p> <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> <a class="link" href="../../main_overview/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">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="Policies">Policy</a><span class="special">></span> <a class="link" href="../../main_overview/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="Policies">Policy</a><span class="special">&);</span> <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> <a class="link" href="../../main_overview/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">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="Policies">Policy</a><span class="special">></span> <a class="link" href="../../main_overview/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="Policies">Policy</a><span class="special">&);</span> </pre> <p> Returns a value <span class="emphasis"><em>x</em></span> such that: <code class="computeroutput"><span class="identifier">p</span> <span class="special">=</span> <span class="identifier">ibeta</span><span class="special">(</span><span class="identifier">a</span><span class="special">,</span> <span class="identifier">b</span><span class="special">,</span> <span class="identifier">x</span><span class="special">);</span></code> and sets <code class="computeroutput"><span class="special">*</span><span class="identifier">py</span> <span class="special">=</span> <span class="number">1</span> <span class="special">-</span> <span class="identifier">x</span></code> when the <code class="computeroutput"><span class="identifier">py</span></code> parameter is provided and is non-null. Note that internally this function computes whichever is 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>, 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 means that even if the function returns <code class="computeroutput"><span class="number">1</span></code>, the value stored in <code class="computeroutput"><span class="special">*</span><span class="identifier">py</span></code> may be non-zero, albeit very small. </p> <p> Requires: <span class="emphasis"><em>a,b > 0</em></span> and <span class="emphasis"><em>0 <= p <= 1</em></span>. </p> <p> </p> <p> The final <a class="link" href="../../policy.html" title="Policies">Policy</a> argument is optional and can be used to control the behaviour of the function: how it handles errors, what level of precision to use etc. Refer to the <a class="link" href="../../policy.html" title="Policies">policy documentation for more details</a>. </p> <p> </p> <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> <a class="link" href="../../main_overview/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">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="Policies">Policy</a><span class="special">></span> <a class="link" href="../../main_overview/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="Policies">Policy</a><span class="special">&);</span> <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> <a class="link" href="../../main_overview/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">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="Policies">Policy</a><span class="special">></span> <a class="link" href="../../main_overview/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="Policies">Policy</a><span class="special">&);</span> </pre> <p> Returns a value <span class="emphasis"><em>x</em></span> such that: <code class="computeroutput"><span class="identifier">q</span> <span class="special">=</span> <span class="identifier">ibetac</span><span class="special">(</span><span class="identifier">a</span><span class="special">,</span> <span class="identifier">b</span><span class="special">,</span> <span class="identifier">x</span><span class="special">);</span></code> and sets <code class="computeroutput"><span class="special">*</span><span class="identifier">py</span> <span class="special">=</span> <span class="number">1</span> <span class="special">-</span> <span class="identifier">x</span></code> when the <code class="computeroutput"><span class="identifier">py</span></code> parameter is provided and is non-null. Note that internally this function computes whichever is 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>, 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 means that even if the function returns <code class="computeroutput"><span class="number">1</span></code>, the value stored in <code class="computeroutput"><span class="special">*</span><span class="identifier">py</span></code> may be non-zero, albeit very small. </p> <p> Requires: <span class="emphasis"><em>a,b > 0</em></span> and <span class="emphasis"><em>0 <= q <= 1</em></span>. </p> <p> </p> <p> The final <a class="link" href="../../policy.html" title="Policies">Policy</a> argument is optional and can be used to control the behaviour of the function: how it handles errors, what level of precision to use etc. Refer to the <a class="link" href="../../policy.html" title="Policies">policy documentation for more details</a>. </p> <p> </p> <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> <a class="link" href="../../main_overview/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">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="Policies">Policy</a><span class="special">></span> <a class="link" href="../../main_overview/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="Policies">Policy</a><span class="special">&);</span> </pre> <p> Returns a value <span class="emphasis"><em>a</em></span> such that: <code class="computeroutput"><span class="identifier">p</span> <span class="special">=</span> <span class="identifier">ibeta</span><span class="special">(</span><span class="identifier">a</span><span class="special">,</span> <span class="identifier">b</span><span class="special">,</span> <span class="identifier">x</span><span class="special">);</span></code> </p> <p> Requires: <span class="emphasis"><em>b > 0</em></span>, <span class="emphasis"><em>0 < x < 1</em></span> and <span class="emphasis"><em>0 <= p <= 1</em></span>. </p> <p> </p> <p> The final <a class="link" href="../../policy.html" title="Policies">Policy</a> argument is optional and can be used to control the behaviour of the function: how it handles errors, what level of precision to use etc. Refer to the <a class="link" href="../../policy.html" title="Policies">policy documentation for more details</a>. </p> <p> </p> <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> <a class="link" href="../../main_overview/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">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="Policies">Policy</a><span class="special">></span> <a class="link" href="../../main_overview/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="Policies">Policy</a><span class="special">&);</span> </pre> <p> Returns a value <span class="emphasis"><em>a</em></span> such that: <code class="computeroutput"><span class="identifier">q</span> <span class="special">=</span> <span class="identifier">ibetac</span><span class="special">(</span><span class="identifier">a</span><span class="special">,</span> <span class="identifier">b</span><span class="special">,</span> <span class="identifier">x</span><span class="special">);</span></code> </p> <p> Requires: <span class="emphasis"><em>b > 0</em></span>, <span class="emphasis"><em>0 < x < 1</em></span> and <span class="emphasis"><em>0 <= q <= 1</em></span>. </p> <p> </p> <p> The final <a class="link" href="../../policy.html" title="Policies">Policy</a> argument is optional and can be used to control the behaviour of the function: how it handles errors, what level of precision to use etc. Refer to the <a class="link" href="../../policy.html" title="Policies">policy documentation for more details</a>. </p> <p> </p> <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> <a class="link" href="../../main_overview/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">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="Policies">Policy</a><span class="special">></span> <a class="link" href="../../main_overview/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="Policies">Policy</a><span class="special">&);</span> </pre> <p> Returns a value <span class="emphasis"><em>b</em></span> such that: <code class="computeroutput"><span class="identifier">p</span> <span class="special">=</span> <span class="identifier">ibeta</span><span class="special">(</span><span class="identifier">a</span><span class="special">,</span> <span class="identifier">b</span><span class="special">,</span> <span class="identifier">x</span><span class="special">);</span></code> </p> <p> Requires: <span class="emphasis"><em>a > 0</em></span>, <span class="emphasis"><em>0 < x < 1</em></span> and <span class="emphasis"><em>0 <= p <= 1</em></span>. </p> <p> </p> <p> The final <a class="link" href="../../policy.html" title="Policies">Policy</a> argument is optional and can be used to control the behaviour of the function: how it handles errors, what level of precision to use etc. Refer to the <a class="link" href="../../policy.html" title="Policies">policy documentation for more details</a>. </p> <p> </p> <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> <a class="link" href="../../main_overview/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">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="Policies">Policy</a><span class="special">></span> <a class="link" href="../../main_overview/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="Policies">Policy</a><span class="special">&);</span> </pre> <p> Returns a value <span class="emphasis"><em>b</em></span> such that: <code class="computeroutput"><span class="identifier">q</span> <span class="special">=</span> <span class="identifier">ibetac</span><span class="special">(</span><span class="identifier">a</span><span class="special">,</span> <span class="identifier">b</span><span class="special">,</span> <span class="identifier">x</span><span class="special">);</span></code> </p> <p> Requires: <span class="emphasis"><em>a > 0</em></span>, <span class="emphasis"><em>0 < x < 1</em></span> and <span class="emphasis"><em>0 <= q <= 1</em></span>. </p> <p> </p> <p> The final <a class="link" href="../../policy.html" title="Policies">Policy</a> argument is optional and can be used to control the behaviour of the function: how it handles errors, what level of precision to use etc. Refer to the <a class="link" href="../../policy.html" title="Policies">policy documentation for more details</a>. </p> <p> </p> <a name="math_toolkit.special.sf_beta.ibeta_inv_function.accuracy"></a><h5> <a name="id1103947"></a> <a class="link" href="ibeta_inv_function.html#math_toolkit.special.sf_beta.ibeta_inv_function.accuracy">Accuracy</a> </h5> <p> The accuracy of these functions should closely follow that of the regular forward incomplete beta functions. However, note that in some parts of their domain, these functions can be extremely sensitive to changes in input, particularly when the argument <span class="emphasis"><em>p</em></span> (or it's complement <span class="emphasis"><em>q</em></span>) is very close to <code class="computeroutput"><span class="number">0</span></code> or <code class="computeroutput"><span class="number">1</span></code>. </p> <a name="math_toolkit.special.sf_beta.ibeta_inv_function.testing"></a><h5> <a name="id1103987"></a> <a class="link" href="ibeta_inv_function.html#math_toolkit.special.sf_beta.ibeta_inv_function.testing">Testing</a> </h5> <p> There are two sets of tests: </p> <div class="itemizedlist"><ul type="disc"> <li> Basic sanity checks attempt to "round-trip" from <span class="emphasis"><em>a, b</em></span> and <span class="emphasis"><em>x</em></span> to <span class="emphasis"><em>p</em></span> or <span class="emphasis"><em>q</em></span> and back again. These tests have quite generous tolerances: in general both the incomplete beta and its inverses change so rapidly, that round tripping to more than a couple of significant digits isn't possible. This is especially true when <span class="emphasis"><em>p</em></span> or <span class="emphasis"><em>q</em></span> is very near one: in this case there isn't enough "information content" in the input to the inverse function to get back where you started. </li> <li> Accuracy checks using high precision test values. These measure the accuracy of the result, given exact input values. </li> </ul></div> <a name="math_toolkit.special.sf_beta.ibeta_inv_function.implementation_of_ibeta_inv_and_ibetac_inv"></a><h5> <a name="id1104044"></a> <a class="link" href="ibeta_inv_function.html#math_toolkit.special.sf_beta.ibeta_inv_function.implementation_of_ibeta_inv_and_ibetac_inv">Implementation of ibeta_inv and ibetac_inv</a> </h5> <p> These two functions share a common implementation. </p> <p> First an initial approximation to x is computed then the last few bits are cleaned up using <a href="http://en.wikipedia.org/wiki/Simple_rational_approximation" target="_top">Halley iteration</a>. The iteration limit is set to 1<span class="emphasis"><em>2 of the number of bits in T, which by experiment is sufficient to ensure that the inverses are at least as accurate as the normal incomplete beta functions. Up to 5 iterations may be required in extreme cases, although normally only one or two are required. Further, the number of iterations required decreases with increasing /a</em></span> and <span class="emphasis"><em>b</em></span> (which generally form the more important use cases). </p> <p> The initial guesses used for iteration are obtained as follows: </p> <p> Firstly recall that: </p> <p> <span class="inlinemediaobject"><img src="../../../../equations/ibeta_inv5.png"></span> </p> <p> We may wish to start from either p or q, and to calculate either x or y. In addition at any stage we can exchange a for b, p for q, and x for y if it results in a more manageable problem. </p> <p> 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 described in: </p> <p> Asymptotic Inversion of the Incomplete Beta Function, by N. M. <a href="http://homepages.cwi.nl/~nicot/" target="_top">Temme</a>. Journal of Computational and Applied Mathematics 41 (1992) 145-157. </p> <p> The nearly symmetrical case (section 2 of the paper) is used for </p> <p> <span class="inlinemediaobject"><img src="../../../../equations/ibeta_inv2.png"></span> </p> <p> and involves solving the inverse error function first. The method is accurate to at least 2 decimal digits when <code class="literal">a = 5</code> rising to at least 8 digits when <code class="literal">a = 10<sup>5</sup></code>. </p> <p> The general error function case (section 3 of the paper) is used for </p> <p> <span class="inlinemediaobject"><img src="../../../../equations/ibeta_inv3.png"></span> </p> <p> and again expresses the inverse incomplete beta in terms of the inverse of the error function. The method is accurate to at least 2 decimal digits when <code class="literal">a+b = 5</code> rising to 11 digits when <code class="literal">a+b = 10<sup>5</sup></code>. However, when the result is expected to be very small, and when a+b is also small, then its accuracy tails off, in this case when p<sup>1/a</sup> < 0.0025 then it is better to use the following as an initial estimate: </p> <p> <span class="inlinemediaobject"><img src="../../../../equations/ibeta_inv4.png"></span> </p> <p> 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> <span class="number">5</span></code> the method of section 4 of the paper is used. This expresses the inverse incomplete beta in terms of the inverse of the incomplete gamma function, and is therefore significantly more expensive to compute than the other cases. However the method is accurate to at least 3 decimal digits when <code class="literal">a = 5</code> rising to at least 10 digits when <code class="literal">a = 10<sup>5</sup></code>. This method is limited to a > b, and therefore we need to perform an exchange a for b, p for q and x for y when this is not the case. In addition when p is close to 1 the method is inaccurate should we actually want y rather than x as output. Therefore when q is small (<code class="literal">q<sup>1/p</sup> < 10<sup>-3</sup></code>) we use: </p> <p> <span class="inlinemediaobject"><img src="../../../../equations/ibeta_inv6.png"></span> </p> <p> which is both cheaper to compute than the full method, and a more accurate estimate on q. </p> <p> When a and b are both small there is a distinct lack of information in the literature on how to proceed. I am extremely grateful to Prof Nico Temme who provided the following information with a great deal of patience and explanation on his part. Any errors that follow are entirely my own, and not Prof Temme's. </p> <p> When a and b are both less than 1, then there is a point of inflection in the incomplete beta at point <code class="computeroutput"><span class="identifier">xs</span> <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> <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> we swap a for b, p for q and x for y, so that now we always look for a point x below the point of inflection <code class="computeroutput"><span class="identifier">xs</span></code>, and on a convex curve. An initial estimate for x is made with: </p> <p> <span class="inlinemediaobject"><img src="../../../../equations/ibeta_inv7.png"></span> </p> <p> which is provably below the true value for x: <a href="http://en.wikipedia.org/wiki/Newton%27s_method" target="_top">Newton iteration</a> will therefore smoothly converge on x without problems caused by overshooting etc. </p> <p> When a and b are both greater than 1, but a+b is too small to use the other methods mentioned above, we proceed as follows. Observe that there is a point of inflection in the incomplete beta at <code class="computeroutput"><span class="identifier">xs</span> <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> <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> we swap a for b, p for q and x for y, so that now we always look for a point x below the point of inflection <code class="computeroutput"><span class="identifier">xs</span></code>, and on a concave curve. An initial estimate for x is made with: </p> <p> <span class="inlinemediaobject"><img src="../../../../equations/ibeta_inv4.png"></span> </p> <p> which can be improved somewhat to: </p> <p> <span class="inlinemediaobject"><img src="../../../../equations/ibeta_inv1.png"></span> </p> <p> when b and x are both small (I've used b < a and x < 0.2). This actually under-estimates x, which drops us on the wrong side of x for Newton iteration to converge monotonically. However, use of higher derivatives and Halley iteration keeps everything under control. </p> <p> The final case to be considered if when one of a and b is less than or equal to 1, and the other greater that 1. Here, if b < a we swap a for b, p for q and x for y. Now the curve of the incomplete beta is convex with no points of inflection in [0,1]. For small p, x can be estimated using </p> <p> <span class="inlinemediaobject"><img src="../../../../equations/ibeta_inv4.png"></span> </p> <p> which under-estimates x, and drops us on the right side of the true value for Newton iteration to converge monotonically. However, when p is large this can quite badly underestimate x. This is especially an issue when we really want to find y, in which case this method can be an arbitrary number of order of magnitudes out, leading to very poor convergence during iteration. </p> <p> Things can be improved by considering the incomplete beta as a distorted quarter circle, and estimating y from: </p> <p> <span class="inlinemediaobject"><img src="../../../../equations/ibeta_inv8.png"></span> </p> <p> This doesn't guarantee that we will drop in on the right side of x for monotonic convergence, but it does get us close enough that Halley iteration rapidly converges on the true value. </p> <a name="math_toolkit.special.sf_beta.ibeta_inv_function.implementation_of_inverses_on_the_a_and_b_parameters"></a><h5> <a name="id1104753"></a> <a class="link" href="ibeta_inv_function.html#math_toolkit.special.sf_beta.ibeta_inv_function.implementation_of_inverses_on_the_a_and_b_parameters">Implementation of inverses on the a and b parameters</a> </h5> <p> These four functions share a common implementation. </p> <p> First an initial approximation is computed for <span class="emphasis"><em>a</em></span> or <span class="emphasis"><em>b</em></span>: where possible this uses a Cornish-Fisher expansion for the negative binomial distribution to get within around 1 of the result. However, when <span class="emphasis"><em>a</em></span> or <span class="emphasis"><em>b</em></span> are very small the Cornish Fisher expansion is not usable, in this case the initial approximation is chosen so that I<sub>x</sub>(a, b) is near the middle of the range [0,1]. </p> <p> This initial guess is then used as a starting value for a generic root finding algorithm. The algorithm converges rapidly on the root once it has been bracketed, but bracketing the root may take several iterations. A better initial approximation for <span class="emphasis"><em>a</em></span> or <span class="emphasis"><em>b</em></span> would improve these functions quite substantially: currently 10-20 incomplete beta function invocations are required to find the root. </p> </div> <table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr> <td align="left"></td> <td align="right"><div class="copyright-footer">Copyright © 2006 , 2007, 2008, 2009 John Maddock, Paul A. Bristow, Hubert Holin, Xiaogang Zhang, Bruno Lalande, Johan Råde, Gautam Sewani and Thijs van den Berg<p> Distributed under the Boost Software License, Version 1.0. (See accompanying file LICENSE_1_0.txt or copy at <a href="http://www.boost.org/LICENSE_1_0.txt" target="_top">http://www.boost.org/LICENSE_1_0.txt</a>) </p> </div></td> </tr></table> <hr> <div class="spirit-nav"> <a accesskey="p" href="ibeta_function.html"><img src="../../../../../../../../doc/src/images/prev.png" alt="Prev"></a><a accesskey="u" href="../sf_beta.html"><img src="../../../../../../../../doc/src/images/up.png" alt="Up"></a><a accesskey="h" href="../../../index.html"><img src="../../../../../../../../doc/src/images/home.png" alt="Home"></a><a accesskey="n" href="beta_derivative.html"><img src="../../../../../../../../doc/src/images/next.png" alt="Next"></a> </div> </body> </html>