| <html> |
| <head> |
| <meta http-equiv="Content-Type" content="text/html; charset=US-ASCII"> |
| <title>Incomplete Beta Functions</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="beta_function.html" title="Beta"> |
| <link rel="next" href="ibeta_inv_function.html" title="The Incomplete Beta Function Inverses"> |
| </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="beta_function.html"><img src="../../../../../../../../doc/src/images/prev.png" alt="Prev"></a><a accesskey="u" href="../sf_beta.html"><img src="../../../../../../../../doc/src/images/up.png" alt="Up"></a><a accesskey="h" href="../../../index.html"><img src="../../../../../../../../doc/src/images/home.png" alt="Home"></a><a accesskey="n" href="ibeta_inv_function.html"><img src="../../../../../../../../doc/src/images/next.png" alt="Next"></a> |
| </div> |
| <div class="section" lang="en"> |
| <div class="titlepage"><div><div><h4 class="title"> |
| <a name="math_toolkit.special.sf_beta.ibeta_function"></a><a class="link" href="ibeta_function.html" title="Incomplete Beta Functions"> Incomplete |
| Beta Functions</a> |
| </h4></div></div></div> |
| <a name="math_toolkit.special.sf_beta.ibeta_function.synopsis"></a><h5> |
| <a name="id1092121"></a> |
| <a class="link" href="ibeta_function.html#math_toolkit.special.sf_beta.ibeta_function.synopsis">Synopsis</a> |
| </h5> |
| <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</span><span class="special">(</span><span class="identifier">T1</span> <span class="identifier">a</span><span class="special">,</span> <span class="identifier">T2</span> <span class="identifier">b</span><span class="special">,</span> <span class="identifier">T3</span> <span class="identifier">x</span><span class="special">);</span> |
| |
| <span class="keyword">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</span><span class="special">(</span><span class="identifier">T1</span> <span class="identifier">a</span><span class="special">,</span> <span class="identifier">T2</span> <span class="identifier">b</span><span class="special">,</span> <span class="identifier">T3</span> <span class="identifier">x</span><span class="special">,</span> <span class="keyword">const</span> <a class="link" href="../../policy.html" title="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</span><span class="special">(</span><span class="identifier">T1</span> <span class="identifier">a</span><span class="special">,</span> <span class="identifier">T2</span> <span class="identifier">b</span><span class="special">,</span> <span class="identifier">T3</span> <span class="identifier">x</span><span class="special">);</span> |
| |
| <span class="keyword">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</span><span class="special">(</span><span class="identifier">T1</span> <span class="identifier">a</span><span class="special">,</span> <span class="identifier">T2</span> <span class="identifier">b</span><span class="special">,</span> <span class="identifier">T3</span> <span class="identifier">x</span><span class="special">,</span> <span class="keyword">const</span> <a class="link" href="../../policy.html" title="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">beta</span><span class="special">(</span><span class="identifier">T1</span> <span class="identifier">a</span><span class="special">,</span> <span class="identifier">T2</span> <span class="identifier">b</span><span class="special">,</span> <span class="identifier">T3</span> <span class="identifier">x</span><span class="special">);</span> |
| |
| <span class="keyword">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">beta</span><span class="special">(</span><span class="identifier">T1</span> <span class="identifier">a</span><span class="special">,</span> <span class="identifier">T2</span> <span class="identifier">b</span><span class="special">,</span> <span class="identifier">T3</span> <span class="identifier">x</span><span class="special">,</span> <span class="keyword">const</span> <a class="link" href="../../policy.html" title="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">betac</span><span class="special">(</span><span class="identifier">T1</span> <span class="identifier">a</span><span class="special">,</span> <span class="identifier">T2</span> <span class="identifier">b</span><span class="special">,</span> <span class="identifier">T3</span> <span class="identifier">x</span><span class="special">);</span> |
| |
| <span class="keyword">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">betac</span><span class="special">(</span><span class="identifier">T1</span> <span class="identifier">a</span><span class="special">,</span> <span class="identifier">T2</span> <span class="identifier">b</span><span class="special">,</span> <span class="identifier">T3</span> <span class="identifier">x</span><span class="special">,</span> <span class="keyword">const</span> <a class="link" href="../../policy.html" title="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_function.description"></a><h5> |
| <a name="id1094852"></a> |
| <a class="link" href="ibeta_function.html#math_toolkit.special.sf_beta.ibeta_function.description">Description</a> |
| </h5> |
| <p> |
| There are four <a href="http://en.wikipedia.org/wiki/Incomplete_beta_function" target="_top">incomplete |
| beta functions</a> : two are normalised versions (also known as <span class="emphasis"><em>regularized</em></span> |
| beta functions) that return values in the range [0, 1], and two are non-normalised |
| and return values in the range [0, <a class="link" href="beta_function.html" title="Beta">beta</a>(a, |
| b)]. Users interested in statistical applications should use the normalised |
| (or <a href="http://mathworld.wolfram.com/RegularizedBetaFunction.html" target="_top">regularized</a> |
| ) versions (ibeta and ibetac). |
| </p> |
| <p> |
| All of these functions require <span class="emphasis"><em>0 <= x <= 1</em></span>. |
| </p> |
| <p> |
| The normalized functions <a class="link" href="ibeta_function.html" title="Incomplete Beta Functions">ibeta</a> |
| and <a class="link" href="ibeta_function.html" title="Incomplete Beta Functions">ibetac</a> |
| require <span class="emphasis"><em>a,b >= 0</em></span>, and in addition that not both |
| <span class="emphasis"><em>a</em></span> and <span class="emphasis"><em>b</em></span> are zero. |
| </p> |
| <p> |
| The functions <a class="link" href="beta_function.html" title="Beta">beta</a> |
| and <a class="link" href="ibeta_function.html" title="Incomplete Beta Functions">betac</a> |
| require <span class="emphasis"><em>a,b > 0</em></span>. |
| </p> |
| <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 T1, T2 and T3 are different |
| types. |
| </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</span><span class="special">(</span><span class="identifier">T1</span> <span class="identifier">a</span><span class="special">,</span> <span class="identifier">T2</span> <span class="identifier">b</span><span class="special">,</span> <span class="identifier">T3</span> <span class="identifier">x</span><span class="special">);</span> |
| |
| <span class="keyword">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</span><span class="special">(</span><span class="identifier">T1</span> <span class="identifier">a</span><span class="special">,</span> <span class="identifier">T2</span> <span class="identifier">b</span><span class="special">,</span> <span class="identifier">T3</span> <span class="identifier">x</span><span class="special">,</span> <span class="keyword">const</span> <a class="link" href="../../policy.html" title="Policies">Policy</a><span class="special">&);</span> |
| </pre> |
| <p> |
| Returns the normalised incomplete beta function of a, b and x: |
| </p> |
| <p> |
| <span class="inlinemediaobject"><img src="../../../../equations/ibeta3.png"></span> |
| </p> |
| <p> |
| <span class="inlinemediaobject"><img src="../../../../graphs/ibeta.png" align="middle"></span> |
| </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</span><span class="special">(</span><span class="identifier">T1</span> <span class="identifier">a</span><span class="special">,</span> <span class="identifier">T2</span> <span class="identifier">b</span><span class="special">,</span> <span class="identifier">T3</span> <span class="identifier">x</span><span class="special">);</span> |
| |
| <span class="keyword">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</span><span class="special">(</span><span class="identifier">T1</span> <span class="identifier">a</span><span class="special">,</span> <span class="identifier">T2</span> <span class="identifier">b</span><span class="special">,</span> <span class="identifier">T3</span> <span class="identifier">x</span><span class="special">,</span> <span class="keyword">const</span> <a class="link" href="../../policy.html" title="Policies">Policy</a><span class="special">&);</span> |
| </pre> |
| <p> |
| Returns the normalised complement of the incomplete beta function of a, |
| b and x: |
| </p> |
| <p> |
| <span class="inlinemediaobject"><img src="../../../../equations/ibeta4.png"></span> |
| </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">beta</span><span class="special">(</span><span class="identifier">T1</span> <span class="identifier">a</span><span class="special">,</span> <span class="identifier">T2</span> <span class="identifier">b</span><span class="special">,</span> <span class="identifier">T3</span> <span class="identifier">x</span><span class="special">);</span> |
| |
| <span class="keyword">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">beta</span><span class="special">(</span><span class="identifier">T1</span> <span class="identifier">a</span><span class="special">,</span> <span class="identifier">T2</span> <span class="identifier">b</span><span class="special">,</span> <span class="identifier">T3</span> <span class="identifier">x</span><span class="special">,</span> <span class="keyword">const</span> <a class="link" href="../../policy.html" title="Policies">Policy</a><span class="special">&);</span> |
| </pre> |
| <p> |
| Returns the full (non-normalised) incomplete beta function of a, b and |
| x: |
| </p> |
| <p> |
| <span class="inlinemediaobject"><img src="../../../../equations/ibeta1.png"></span> |
| </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">betac</span><span class="special">(</span><span class="identifier">T1</span> <span class="identifier">a</span><span class="special">,</span> <span class="identifier">T2</span> <span class="identifier">b</span><span class="special">,</span> <span class="identifier">T3</span> <span class="identifier">x</span><span class="special">);</span> |
| |
| <span class="keyword">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">betac</span><span class="special">(</span><span class="identifier">T1</span> <span class="identifier">a</span><span class="special">,</span> <span class="identifier">T2</span> <span class="identifier">b</span><span class="special">,</span> <span class="identifier">T3</span> <span class="identifier">x</span><span class="special">,</span> <span class="keyword">const</span> <a class="link" href="../../policy.html" title="Policies">Policy</a><span class="special">&);</span> |
| </pre> |
| <p> |
| Returns the full (non-normalised) complement of the incomplete beta function |
| of a, b and x: |
| </p> |
| <p> |
| <span class="inlinemediaobject"><img src="../../../../equations/ibeta2.png"></span> |
| </p> |
| <a name="math_toolkit.special.sf_beta.ibeta_function.accuracy"></a><h5> |
| <a name="id1096026"></a> |
| <a class="link" href="ibeta_function.html#math_toolkit.special.sf_beta.ibeta_function.accuracy">Accuracy</a> |
| </h5> |
| <p> |
| The following tables give peak and mean relative errors in over various |
| domains of a, b and x, along with comparisons to the <a href="http://www.gnu.org/software/gsl/" target="_top">GSL-1.9</a> |
| and <a href="http://www.netlib.org/cephes/" target="_top">Cephes</a> libraries. |
| Note that only results for the widest floating-point type on the system |
| are given as narrower types have <a class="link" href="../../backgrounders/relative_error.html#zero_error">effectively |
| zero error</a>. |
| </p> |
| <p> |
| Note that the results for 80 and 128-bit long doubles are noticeably higher |
| than for doubles: this is because the wider exponent range of these types |
| allow more extreme test cases to be tested. For example expected results |
| that are zero at double precision, may be finite but exceptionally small |
| with the wider exponent range of the long double types. |
| </p> |
| <div class="table"> |
| <a name="math_toolkit.special.sf_beta.ibeta_function.errors_in_the_function_ibeta_a_b_x_"></a><p class="title"><b>Table 23. Errors In the Function ibeta(a,b,x)</b></p> |
| <div class="table-contents"><table class="table" summary="Errors In the Function ibeta(a,b,x)"> |
| <colgroup> |
| <col> |
| <col> |
| <col> |
| <col> |
| <col> |
| </colgroup> |
| <thead><tr> |
| <th> |
| <p> |
| Significand Size |
| </p> |
| </th> |
| <th> |
| <p> |
| Platform and Compiler |
| </p> |
| </th> |
| <th> |
| <p> |
| 0 < a,b < 10 |
| </p> |
| <p> |
| and |
| </p> |
| <p> |
| 0 < x < 1 |
| </p> |
| </th> |
| <th> |
| <p> |
| 0 < a,b < 100 |
| </p> |
| <p> |
| and |
| </p> |
| <p> |
| 0 < x < 1 |
| </p> |
| </th> |
| <th> |
| <p> |
| 1x10<sup>-5</sup> < a,b < 1x10<sup>5</sup> |
| </p> |
| <p> |
| and |
| </p> |
| <p> |
| 0 < x < 1 |
| </p> |
| </th> |
| </tr></thead> |
| <tbody> |
| <tr> |
| <td> |
| <p> |
| 53 |
| </p> |
| </td> |
| <td> |
| <p> |
| Win32, Visual C++ 8 |
| </p> |
| </td> |
| <td> |
| <p> |
| Peak=42.3 Mean=2.9 |
| </p> |
| <p> |
| (GSL Peak=682 Mean=32.5) |
| </p> |
| <p> |
| (<a href="http://www.netlib.org/cephes/" target="_top">Cephes</a> Peak=42.7 |
| Mean=7.0) |
| </p> |
| </td> |
| <td> |
| <p> |
| Peak=108 Mean=16.6 |
| </p> |
| <p> |
| (GSL Peak=690 Mean=151) |
| </p> |
| <p> |
| (<a href="http://www.netlib.org/cephes/" target="_top">Cephes</a> Peak=1545 |
| Mean=218) |
| </p> |
| </td> |
| <td> |
| <p> |
| Peak=4x10<sup>3</sup> ​ Mean=203 |
| </p> |
| <p> |
| (GSL Peak~3x10<sup>5</sup> ​ Mean~2x10<sup>4</sup> ​) |
| </p> |
| <p> |
| (<a href="http://www.netlib.org/cephes/" target="_top">Cephes</a> Peak~5x10<sup>5</sup> ​ Mean~2x10<sup>4</sup> ​) |
| </p> |
| </td> |
| </tr> |
| <tr> |
| <td> |
| <p> |
| 64 |
| </p> |
| </td> |
| <td> |
| <p> |
| Redhat Linux IA32, gcc-3.4.4 |
| </p> |
| </td> |
| <td> |
| <p> |
| Peak=21.9 Mean=3.1 |
| </p> |
| </td> |
| <td> |
| <p> |
| Peak=270.7 Mean=26.8 |
| </p> |
| </td> |
| <td> |
| <p> |
| Peak~5x10<sup>4</sup> ​ Mean=3x10<sup>3</sup> ​ |
| </p> |
| </td> |
| </tr> |
| <tr> |
| <td> |
| <p> |
| 64 |
| </p> |
| </td> |
| <td> |
| <p> |
| Redhat Linux IA64, gcc-3.4.4 |
| </p> |
| </td> |
| <td> |
| <p> |
| Peak=15.4 Mean=3.0 |
| </p> |
| </td> |
| <td> |
| <p> |
| Peak=112.9 Mean=14.3 |
| </p> |
| </td> |
| <td> |
| <p> |
| Peak~5x10<sup>4</sup> ​ Mean=3x10<sup>3</sup> ​ |
| </p> |
| </td> |
| </tr> |
| <tr> |
| <td> |
| <p> |
| 113 |
| </p> |
| </td> |
| <td> |
| <p> |
| HPUX IA64, aCC A.06.06 |
| </p> |
| </td> |
| <td> |
| <p> |
| Peak=20.9 Mean=2.6 |
| </p> |
| </td> |
| <td> |
| <p> |
| Peak=88.1 Mean=14.3 |
| </p> |
| </td> |
| <td> |
| <p> |
| Peak~2x10<sup>4</sup> ​ Mean=1x10<sup>3</sup> ​ |
| </p> |
| </td> |
| </tr> |
| </tbody> |
| </table></div> |
| </div> |
| <br class="table-break"><div class="table"> |
| <a name="math_toolkit.special.sf_beta.ibeta_function.errors_in_the_function_ibetac_a_b_x_"></a><p class="title"><b>Table 24. Errors In the Function ibetac(a,b,x)</b></p> |
| <div class="table-contents"><table class="table" summary="Errors In the Function ibetac(a,b,x)"> |
| <colgroup> |
| <col> |
| <col> |
| <col> |
| <col> |
| <col> |
| </colgroup> |
| <thead><tr> |
| <th> |
| <p> |
| Significand Size |
| </p> |
| </th> |
| <th> |
| <p> |
| Platform and Compiler |
| </p> |
| </th> |
| <th> |
| <p> |
| 0 < a,b < 10 |
| </p> |
| <p> |
| and |
| </p> |
| <p> |
| 0 < x < 1 |
| </p> |
| </th> |
| <th> |
| <p> |
| 0 < a,b < 100 |
| </p> |
| <p> |
| and |
| </p> |
| <p> |
| 0 < x < 1 |
| </p> |
| </th> |
| <th> |
| <p> |
| 1x10<sup>-5</sup> < a,b < 1x10<sup>5</sup> |
| </p> |
| <p> |
| and |
| </p> |
| <p> |
| 0 < x < 1 |
| </p> |
| </th> |
| </tr></thead> |
| <tbody> |
| <tr> |
| <td> |
| <p> |
| 53 |
| </p> |
| </td> |
| <td> |
| <p> |
| Win32, Visual C++ 8 |
| </p> |
| </td> |
| <td> |
| <p> |
| Peak=13.9 Mean=2.0 |
| </p> |
| </td> |
| <td> |
| <p> |
| Peak=56.2 Mean=14 |
| </p> |
| </td> |
| <td> |
| <p> |
| Peak=3x10<sup>3</sup> ​ Mean=159 |
| </p> |
| </td> |
| </tr> |
| <tr> |
| <td> |
| <p> |
| 64 |
| </p> |
| </td> |
| <td> |
| <p> |
| Redhat Linux IA32, gcc-3.4.4 |
| </p> |
| </td> |
| <td> |
| <p> |
| Peak=21.1 Mean=3.6 |
| </p> |
| </td> |
| <td> |
| <p> |
| Peak=221.7 Mean=25.8 |
| </p> |
| </td> |
| <td> |
| <p> |
| Peak~9x10<sup>4</sup> ​ Mean=3x10<sup>3</sup> ​ |
| </p> |
| </td> |
| </tr> |
| <tr> |
| <td> |
| <p> |
| 64 |
| </p> |
| </td> |
| <td> |
| <p> |
| Redhat Linux IA64, gcc-3.4.4 |
| </p> |
| </td> |
| <td> |
| <p> |
| Peak=10.6 Mean=2.2 |
| </p> |
| </td> |
| <td> |
| <p> |
| Peak=73.9 Mean=11.9 |
| </p> |
| </td> |
| <td> |
| <p> |
| Peak~9x10<sup>4</sup> ​ Mean=3x10<sup>3</sup> ​ |
| </p> |
| </td> |
| </tr> |
| <tr> |
| <td> |
| <p> |
| 113 |
| </p> |
| </td> |
| <td> |
| <p> |
| HPUX IA64, aCC A.06.06 |
| </p> |
| </td> |
| <td> |
| <p> |
| Peak=9.9 Mean=2.6 |
| </p> |
| </td> |
| <td> |
| <p> |
| Peak=117.7 Mean=15.1 |
| </p> |
| </td> |
| <td> |
| <p> |
| Peak~3x10<sup>4</sup> ​ Mean=1x10<sup>3</sup> ​ |
| </p> |
| </td> |
| </tr> |
| </tbody> |
| </table></div> |
| </div> |
| <br class="table-break"><div class="table"> |
| <a name="math_toolkit.special.sf_beta.ibeta_function.errors_in_the_function_beta_a__b__x_"></a><p class="title"><b>Table 25. Errors In the Function beta(a, b, x)</b></p> |
| <div class="table-contents"><table class="table" summary="Errors In the Function beta(a, b, x)"> |
| <colgroup> |
| <col> |
| <col> |
| <col> |
| <col> |
| <col> |
| </colgroup> |
| <thead><tr> |
| <th> |
| <p> |
| Significand Size |
| </p> |
| </th> |
| <th> |
| <p> |
| Platform and Compiler |
| </p> |
| </th> |
| <th> |
| <p> |
| 0 < a,b < 10 |
| </p> |
| <p> |
| and |
| </p> |
| <p> |
| 0 < x < 1 |
| </p> |
| </th> |
| <th> |
| <p> |
| 0 < a,b < 100 |
| </p> |
| <p> |
| and |
| </p> |
| <p> |
| 0 < x < 1 |
| </p> |
| </th> |
| <th> |
| <p> |
| 1x10<sup>-5</sup> < a,b < 1x10<sup>5</sup> |
| </p> |
| <p> |
| and |
| </p> |
| <p> |
| 0 < x < 1 |
| </p> |
| </th> |
| </tr></thead> |
| <tbody> |
| <tr> |
| <td> |
| <p> |
| 53 |
| </p> |
| </td> |
| <td> |
| <p> |
| Win32, Visual C++ 8 |
| </p> |
| </td> |
| <td> |
| <p> |
| Peak=39 Mean=2.9 |
| </p> |
| </td> |
| <td> |
| <p> |
| Peak=91 Mean=12.7 |
| </p> |
| </td> |
| <td> |
| <p> |
| Peak=635 Mean=25 |
| </p> |
| </td> |
| </tr> |
| <tr> |
| <td> |
| <p> |
| 64 |
| </p> |
| </td> |
| <td> |
| <p> |
| Redhat Linux IA32, gcc-3.4.4 |
| </p> |
| </td> |
| <td> |
| <p> |
| Peak=26 Mean=3.6 |
| </p> |
| </td> |
| <td> |
| <p> |
| Peak=180.7 Mean=30.1 |
| </p> |
| </td> |
| <td> |
| <p> |
| Peak~7x10<sup>4</sup> ​ Mean=3x10<sup>3</sup> ​ |
| </p> |
| </td> |
| </tr> |
| <tr> |
| <td> |
| <p> |
| 64 |
| </p> |
| </td> |
| <td> |
| <p> |
| Redhat Linux IA64, gcc-3.4.4 |
| </p> |
| </td> |
| <td> |
| <p> |
| Peak=13 Mean=2.4 |
| </p> |
| </td> |
| <td> |
| <p> |
| Peak=67.1 Mean=13.4 |
| </p> |
| </td> |
| <td> |
| <p> |
| Peak~7x10<sup>4</sup> ​ Mean=3x10<sup>3</sup> ​ |
| </p> |
| </td> |
| </tr> |
| <tr> |
| <td> |
| <p> |
| 113 |
| </p> |
| </td> |
| <td> |
| <p> |
| HPUX IA64, aCC A.06.06 |
| </p> |
| </td> |
| <td> |
| <p> |
| Peak=27.3 Mean=3.6 |
| </p> |
| </td> |
| <td> |
| <p> |
| Peak=49.8 Mean=9.1 |
| </p> |
| </td> |
| <td> |
| <p> |
| Peak~6x10<sup>4</sup> ​ Mean=3x10<sup>3</sup> ​ |
| </p> |
| </td> |
| </tr> |
| </tbody> |
| </table></div> |
| </div> |
| <br class="table-break"><div class="table"> |
| <a name="math_toolkit.special.sf_beta.ibeta_function.errors_in_the_function_betac_a_b_x_"></a><p class="title"><b>Table 26. Errors In the Function betac(a,b,x)</b></p> |
| <div class="table-contents"><table class="table" summary="Errors In the Function betac(a,b,x)"> |
| <colgroup> |
| <col> |
| <col> |
| <col> |
| <col> |
| <col> |
| </colgroup> |
| <thead><tr> |
| <th> |
| <p> |
| Significand Size |
| </p> |
| </th> |
| <th> |
| <p> |
| Platform and Compiler |
| </p> |
| </th> |
| <th> |
| <p> |
| 0 < a,b < 10 |
| </p> |
| <p> |
| and |
| </p> |
| <p> |
| 0 < x < 1 |
| </p> |
| </th> |
| <th> |
| <p> |
| 0 < a,b < 100 |
| </p> |
| <p> |
| and |
| </p> |
| <p> |
| 0 < x < 1 |
| </p> |
| </th> |
| <th> |
| <p> |
| 1x10<sup>-5</sup> < a,b < 1x10<sup>5</sup> |
| </p> |
| <p> |
| and |
| </p> |
| <p> |
| 0 < x < 1 |
| </p> |
| </th> |
| </tr></thead> |
| <tbody> |
| <tr> |
| <td> |
| <p> |
| 53 |
| </p> |
| </td> |
| <td> |
| <p> |
| Win32, Visual C++ 8 |
| </p> |
| </td> |
| <td> |
| <p> |
| Peak=12.0 Mean=2.4 |
| </p> |
| </td> |
| <td> |
| <p> |
| Peak=91 Mean=15 |
| </p> |
| </td> |
| <td> |
| <p> |
| Peak=4x10<sup>3</sup> ​ Mean=113 |
| </p> |
| </td> |
| </tr> |
| <tr> |
| <td> |
| <p> |
| 64 |
| </p> |
| </td> |
| <td> |
| <p> |
| Redhat Linux IA32, gcc-3.4.4 |
| </p> |
| </td> |
| <td> |
| <p> |
| Peak=19.8 Mean=3.8 |
| </p> |
| </td> |
| <td> |
| <p> |
| Peak=295.1 Mean=33.9 |
| </p> |
| </td> |
| <td> |
| <p> |
| Peak~1x10<sup>5</sup> ​ Mean=5x10<sup>3</sup> ​ |
| </p> |
| </td> |
| </tr> |
| <tr> |
| <td> |
| <p> |
| 64 |
| </p> |
| </td> |
| <td> |
| <p> |
| Redhat Linux IA64, gcc-3.4.4 |
| </p> |
| </td> |
| <td> |
| <p> |
| Peak=11.2 Mean=2.4 |
| </p> |
| </td> |
| <td> |
| <p> |
| Peak=63.5 Mean=13.6 |
| </p> |
| </td> |
| <td> |
| <p> |
| Peak~1x10<sup>5</sup> ​ Mean=5x10<sup>3</sup> ​ |
| </p> |
| </td> |
| </tr> |
| <tr> |
| <td> |
| <p> |
| 113 |
| </p> |
| </td> |
| <td> |
| <p> |
| HPUX IA64, aCC A.06.06 |
| </p> |
| </td> |
| <td> |
| <p> |
| Peak=15.6 Mean=3.5 |
| </p> |
| </td> |
| <td> |
| <p> |
| Peak=39.8 Mean=8.9 |
| </p> |
| </td> |
| <td> |
| <p> |
| Peak~9x10<sup>4</sup> ​ Mean=5x10<sup>3</sup> ​ |
| </p> |
| </td> |
| </tr> |
| </tbody> |
| </table></div> |
| </div> |
| <br class="table-break"><a name="math_toolkit.special.sf_beta.ibeta_function.testing"></a><h5> |
| <a name="id1097271"></a> |
| <a class="link" href="ibeta_function.html#math_toolkit.special.sf_beta.ibeta_function.testing">Testing</a> |
| </h5> |
| <p> |
| There are two sets of tests: spot tests compare values taken from <a href="http://functions.wolfram.com/webMathematica/FunctionEvaluation.jsp?name=BetaRegularized" target="_top">Mathworld's |
| online function evaluator</a> with this implementation: they provide |
| a basic "sanity check" for the implementation, with one spot-test |
| in each implementation-domain (see implementation notes below). |
| </p> |
| <p> |
| Accuracy tests use data generated at very high precision (with <a href="http://shoup.net/ntl/doc/RR.txt" target="_top">NTL |
| RR class</a> set at 1000-bit precision), using the "textbook" |
| continued fraction representation (refer to the first continued fraction |
| in the implementation discussion below). Note that this continued fraction |
| is <span class="emphasis"><em>not</em></span> used in the implementation, and therefore we |
| have test data that is fully independent of the code. |
| </p> |
| <a name="math_toolkit.special.sf_beta.ibeta_function.implementation"></a><h5> |
| <a name="id1097307"></a> |
| <a class="link" href="ibeta_function.html#math_toolkit.special.sf_beta.ibeta_function.implementation">Implementation</a> |
| </h5> |
| <p> |
| This implementation is closely based upon <a href="http://portal.acm.org/citation.cfm?doid=131766.131776" target="_top">"Algorithm |
| 708; Significant digit computation of the incomplete beta function ratios", |
| DiDonato and Morris, ACM, 1992.</a> |
| </p> |
| <p> |
| All four of these functions share a common implementation: this is passed |
| both x and y, and can return either p or q where these are related by: |
| </p> |
| <p> |
| <span class="inlinemediaobject"><img src="../../../../equations/ibeta_inv5.png"></span> |
| </p> |
| <p> |
| so at any point we can swap a for b, x for y and p for q if this results |
| in a more favourable position. Generally such swaps are performed so that |
| we always compute a value less than 0.9: when required this can then be |
| subtracted from 1 without undue cancellation error. |
| </p> |
| <p> |
| The following continued fraction representation is found in many textbooks |
| but is not used in this implementation - it's both slower and less accurate |
| than the alternatives - however it is used to generate test data: |
| </p> |
| <p> |
| <span class="inlinemediaobject"><img src="../../../../equations/ibeta5.png"></span> |
| </p> |
| <p> |
| The following continued fraction is due to <a href="http://portal.acm.org/citation.cfm?doid=131766.131776" target="_top">Didonato |
| and Morris</a>, and is used in this implementation when a and b are |
| both greater than 1: |
| </p> |
| <p> |
| <span class="inlinemediaobject"><img src="../../../../equations/ibeta6.png"></span> |
| </p> |
| <p> |
| For smallish b and x then a series representation can be used: |
| </p> |
| <p> |
| <span class="inlinemediaobject"><img src="../../../../equations/ibeta7.png"></span> |
| </p> |
| <p> |
| When b << a then the transition from 0 to 1 occurs very close to |
| x = 1 and some care has to be taken over the method of computation, in |
| that case the following series representation is used: |
| </p> |
| <p> |
| <span class="inlinemediaobject"><img src="../../../../equations/ibeta8.png"></span> |
| |
| </p> |
| <p> |
| Where Q(a,x) is an <a href="http://functions.wolfram.com/GammaBetaErf/Gamma2/" target="_top">incomplete |
| gamma function</a>. Note that this method relies on keeping a table |
| of all the p<sub>n </sub> previously computed, which does limit the precision of the |
| method, depending upon the size of the table used. |
| </p> |
| <p> |
| When <span class="emphasis"><em>a</em></span> and <span class="emphasis"><em>b</em></span> are both small integers, |
| then we can relate the incomplete beta to the binomial distribution and |
| use the following finite sum: |
| </p> |
| <p> |
| <span class="inlinemediaobject"><img src="../../../../equations/ibeta12.png"></span> |
| </p> |
| <p> |
| Finally we can sidestep difficult areas, or move to an area with a more |
| efficient means of computation, by using the duplication formulae: |
| </p> |
| <p> |
| <span class="inlinemediaobject"><img src="../../../../equations/ibeta10.png"></span> |
| </p> |
| <p> |
| <span class="inlinemediaobject"><img src="../../../../equations/ibeta11.png"></span> |
| </p> |
| <p> |
| The domains of a, b and x for which the various methods are used are identical |
| to those described in the <a href="http://portal.acm.org/citation.cfm?doid=131766.131776" target="_top">Didonato |
| and Morris TOMS 708 paper</a>. |
| </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, 2010 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="beta_function.html"><img src="../../../../../../../../doc/src/images/prev.png" alt="Prev"></a><a accesskey="u" href="../sf_beta.html"><img src="../../../../../../../../doc/src/images/up.png" alt="Up"></a><a accesskey="h" href="../../../index.html"><img src="../../../../../../../../doc/src/images/home.png" alt="Home"></a><a accesskey="n" href="ibeta_inv_function.html"><img src="../../../../../../../../doc/src/images/next.png" alt="Next"></a> |
| </div> |
| </body> |
| </html> |
