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<a accesskey="p" href="ibeta_inv_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="../sf_erf.html"><img src="../../../../../../doc/src/images/next.png" alt="Next"></a>
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<div class="section">
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<div class="titlepage"><div><div><h3 class="title">
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<a name="math_toolkit.sf_beta.beta_derivative"></a><a class="link" href="beta_derivative.html" title="Derivative of the Incomplete Beta Function">Derivative of the
|
||||
Incomplete Beta Function</a>
|
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</h3></div></div></div>
|
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<h5>
|
||||
<a name="math_toolkit.sf_beta.beta_derivative.h0"></a>
|
||||
<span class="phrase"><a name="math_toolkit.sf_beta.beta_derivative.synopsis"></a></span><a class="link" href="beta_derivative.html#math_toolkit.sf_beta.beta_derivative.synopsis">Synopsis</a>
|
||||
</h5>
|
||||
<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>
|
||||
<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="../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_derivative</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="Chapter 15. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">></span>
|
||||
<a class="link" href="../result_type.html" title="Calculation of the Type of the Result"><span class="emphasis"><em>calculated-result-type</em></span></a> <span class="identifier">ibeta_derivative</span><span class="special">(</span><span class="identifier">T1</span> <span class="identifier">a</span><span class="special">,</span> <span class="identifier">T2</span> <span class="identifier">b</span><span class="special">,</span> <span class="identifier">T3</span> <span class="identifier">x</span><span class="special">,</span> <span class="keyword">const</span> <a class="link" href="../../policy.html" title="Chapter 15. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&);</span>
|
||||
|
||||
<span class="special">}}</span> <span class="comment">// namespaces</span>
|
||||
</pre>
|
||||
<h5>
|
||||
<a name="math_toolkit.sf_beta.beta_derivative.h1"></a>
|
||||
<span class="phrase"><a name="math_toolkit.sf_beta.beta_derivative.description"></a></span><a class="link" href="beta_derivative.html#math_toolkit.sf_beta.beta_derivative.description">Description</a>
|
||||
</h5>
|
||||
<p>
|
||||
This function finds some uses in statistical distributions: it computes the
|
||||
partial derivative with respect to <span class="emphasis"><em>x</em></span> of the incomplete
|
||||
beta function <a class="link" href="ibeta_function.html" title="Incomplete Beta Functions">ibeta</a>.
|
||||
</p>
|
||||
<p>
|
||||
<span class="inlinemediaobject"><img src="../../../equations/derivative2.svg"></span>
|
||||
</p>
|
||||
<p>
|
||||
The return type of this function is computed using the <a class="link" href="../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>
|
||||
The final <a class="link" href="../../policy.html" title="Chapter 15. Policies: Controlling Precision, Error Handling etc">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="Chapter 15. Policies: Controlling Precision, Error Handling etc">policy
|
||||
documentation for more details</a>.
|
||||
</p>
|
||||
<h5>
|
||||
<a name="math_toolkit.sf_beta.beta_derivative.h2"></a>
|
||||
<span class="phrase"><a name="math_toolkit.sf_beta.beta_derivative.accuracy"></a></span><a class="link" href="beta_derivative.html#math_toolkit.sf_beta.beta_derivative.accuracy">Accuracy</a>
|
||||
</h5>
|
||||
<p>
|
||||
Almost identical to the incomplete beta function <a class="link" href="ibeta_function.html" title="Incomplete Beta Functions">ibeta</a>.
|
||||
</p>
|
||||
<h5>
|
||||
<a name="math_toolkit.sf_beta.beta_derivative.h3"></a>
|
||||
<span class="phrase"><a name="math_toolkit.sf_beta.beta_derivative.implementation"></a></span><a class="link" href="beta_derivative.html#math_toolkit.sf_beta.beta_derivative.implementation">Implementation</a>
|
||||
</h5>
|
||||
<p>
|
||||
This function just expose some of the internals of the incomplete beta function
|
||||
<a class="link" href="ibeta_function.html" title="Incomplete Beta Functions">ibeta</a>: refer to
|
||||
the documentation for that function for more information.
|
||||
</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 © 2006-2010, 2012-2014 Nikhar Agrawal,
|
||||
Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos, Hubert
|
||||
Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Johan Råde, Gautam Sewani,
|
||||
Benjamin Sobotta, Thijs van den Berg, Daryle Walker and Xiaogang Zhang<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>
|
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<div class="spirit-nav">
|
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<a accesskey="p" href="ibeta_inv_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="../sf_erf.html"><img src="../../../../../../doc/src/images/next.png" alt="Next"></a>
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</body>
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</html>
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@@ -0,0 +1,318 @@
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<html>
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<head>
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<meta http-equiv="Content-Type" content="text/html; charset=US-ASCII">
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<title>Beta</title>
|
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|
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|
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<hr>
|
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<div class="spirit-nav">
|
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<a accesskey="p" href="../sf_beta.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_function.html"><img src="../../../../../../doc/src/images/next.png" alt="Next"></a>
|
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</div>
|
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<div class="section">
|
||||
<div class="titlepage"><div><div><h3 class="title">
|
||||
<a name="math_toolkit.sf_beta.beta_function"></a><a class="link" href="beta_function.html" title="Beta">Beta</a>
|
||||
</h3></div></div></div>
|
||||
<h5>
|
||||
<a name="math_toolkit.sf_beta.beta_function.h0"></a>
|
||||
<span class="phrase"><a name="math_toolkit.sf_beta.beta_function.synopsis"></a></span><a class="link" href="beta_function.html#math_toolkit.sf_beta.beta_function.synopsis">Synopsis</a>
|
||||
</h5>
|
||||
<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>
|
||||
<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>
|
||||
<a class="link" href="../result_type.html" title="Calculation of the Type of the Result"><span class="emphasis"><em>calculated-result-type</em></span></a> <span class="identifier">beta</span><span class="special">(</span><span class="identifier">T1</span> <span class="identifier">a</span><span class="special">,</span> <span class="identifier">T2</span> <span class="identifier">b</span><span class="special">);</span>
|
||||
|
||||
<span class="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> <a class="link" href="../../policy.html" title="Chapter 15. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">></span>
|
||||
<a class="link" href="../result_type.html" title="Calculation of the Type of the Result"><span class="emphasis"><em>calculated-result-type</em></span></a> <span class="identifier">beta</span><span class="special">(</span><span class="identifier">T1</span> <span class="identifier">a</span><span class="special">,</span> <span class="identifier">T2</span> <span class="identifier">b</span><span class="special">,</span> <span class="keyword">const</span> <a class="link" href="../../policy.html" title="Chapter 15. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&);</span>
|
||||
|
||||
<span class="special">}}</span> <span class="comment">// namespaces</span>
|
||||
</pre>
|
||||
<h5>
|
||||
<a name="math_toolkit.sf_beta.beta_function.h1"></a>
|
||||
<span class="phrase"><a name="math_toolkit.sf_beta.beta_function.description"></a></span><a class="link" href="beta_function.html#math_toolkit.sf_beta.beta_function.description">Description</a>
|
||||
</h5>
|
||||
<p>
|
||||
The beta function is defined by:
|
||||
</p>
|
||||
<p>
|
||||
<span class="inlinemediaobject"><img src="../../../equations/beta1.svg"></span>
|
||||
</p>
|
||||
<p>
|
||||
<span class="inlinemediaobject"><img src="../../../graphs/beta.svg" align="middle"></span>
|
||||
</p>
|
||||
<p>
|
||||
The final <a class="link" href="../../policy.html" title="Chapter 15. Policies: Controlling Precision, Error Handling etc">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="Chapter 15. Policies: Controlling Precision, Error Handling etc">policy
|
||||
documentation for more details</a>.
|
||||
</p>
|
||||
<p>
|
||||
There are effectively two versions of this function internally: a fully generic
|
||||
version that is slow, but reasonably accurate, and a much more efficient
|
||||
approximation that is used where the number of digits in the significand
|
||||
of T correspond to a certain <a class="link" href="../lanczos.html" title="The Lanczos Approximation">Lanczos
|
||||
approximation</a>. In practice any built-in floating-point type you will
|
||||
encounter has an appropriate <a class="link" href="../lanczos.html" title="The Lanczos Approximation">Lanczos
|
||||
approximation</a> defined for it. It is also possible, given enough machine
|
||||
time, to generate further <a class="link" href="../lanczos.html" title="The Lanczos Approximation">Lanczos approximation</a>'s
|
||||
using the program libs/math/tools/lanczos_generator.cpp.
|
||||
</p>
|
||||
<p>
|
||||
The return type of these functions is computed using the <a class="link" href="../result_type.html" title="Calculation of the Type of the Result"><span class="emphasis"><em>result
|
||||
type calculation rules</em></span></a> when T1 and T2 are different types.
|
||||
</p>
|
||||
<h5>
|
||||
<a name="math_toolkit.sf_beta.beta_function.h2"></a>
|
||||
<span class="phrase"><a name="math_toolkit.sf_beta.beta_function.accuracy"></a></span><a class="link" href="beta_function.html#math_toolkit.sf_beta.beta_function.accuracy">Accuracy</a>
|
||||
</h5>
|
||||
<p>
|
||||
The following table shows peak errors for various domains of input arguments,
|
||||
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="../relative_error.html#math_toolkit.relative_error.zero_error">effectively
|
||||
zero error</a>.
|
||||
</p>
|
||||
<div class="table">
|
||||
<a name="math_toolkit.sf_beta.beta_function.table_beta"></a><p class="title"><b>Table 6.17. Error rates for beta</b></p>
|
||||
<div class="table-contents"><table class="table" summary="Error rates for beta">
|
||||
<colgroup>
|
||||
<col>
|
||||
<col>
|
||||
<col>
|
||||
<col>
|
||||
<col>
|
||||
</colgroup>
|
||||
<thead><tr>
|
||||
<th>
|
||||
</th>
|
||||
<th>
|
||||
<p>
|
||||
Microsoft Visual C++ version 12.0<br> Win32<br> double
|
||||
</p>
|
||||
</th>
|
||||
<th>
|
||||
<p>
|
||||
GNU C++ version 5.1.0<br> linux<br> long double
|
||||
</p>
|
||||
</th>
|
||||
<th>
|
||||
<p>
|
||||
GNU C++ version 5.1.0<br> linux<br> double
|
||||
</p>
|
||||
</th>
|
||||
<th>
|
||||
<p>
|
||||
Sun compiler version 0x5130<br> Sun Solaris<br> long double
|
||||
</p>
|
||||
</th>
|
||||
</tr></thead>
|
||||
<tbody>
|
||||
<tr>
|
||||
<td>
|
||||
<p>
|
||||
Beta Function: Small Values
|
||||
</p>
|
||||
</td>
|
||||
<td>
|
||||
<p>
|
||||
<span class="blue">Max = 1.75ε (Mean = 0.828ε)</span>
|
||||
</p>
|
||||
</td>
|
||||
<td>
|
||||
<p>
|
||||
<span class="blue">Max = 2.65ε (Mean = 1.22ε)</span><br> <br>
|
||||
(<span class="emphasis"><em><tr1/cmath>:</em></span> Max = 11.9ε (Mean = 7.03ε))
|
||||
</p>
|
||||
</td>
|
||||
<td>
|
||||
<p>
|
||||
<span class="blue">Max = 0ε (Mean = 0ε)</span><br> <br> (<span class="emphasis"><em>GSL
|
||||
1.16:</em></span> Max = 3.94ε (Mean = 1.44ε))<br> (<span class="emphasis"><em>Rmath
|
||||
3.0.2:</em></span> Max = 1.14ε (Mean = 0.601ε))<br> (<span class="emphasis"><em>Cephes:</em></span>
|
||||
Max = 1.97ε (Mean = 0.934ε))
|
||||
</p>
|
||||
</td>
|
||||
<td>
|
||||
<p>
|
||||
<span class="blue">Max = 2.86ε (Mean = 1.31ε)</span>
|
||||
</p>
|
||||
</td>
|
||||
</tr>
|
||||
<tr>
|
||||
<td>
|
||||
<p>
|
||||
Beta Function: Medium Values
|
||||
</p>
|
||||
</td>
|
||||
<td>
|
||||
<p>
|
||||
<span class="blue">Max = 99.1ε (Mean = 22.4ε)</span>
|
||||
</p>
|
||||
</td>
|
||||
<td>
|
||||
<p>
|
||||
<span class="blue">Max = 60.4ε (Mean = 19.4ε)</span><br> <br>
|
||||
(<span class="emphasis"><em><tr1/cmath>:</em></span> Max = 1.07e+03ε (Mean =
|
||||
264ε))
|
||||
</p>
|
||||
</td>
|
||||
<td>
|
||||
<p>
|
||||
<span class="blue">Max = 0.978ε (Mean = 0.0578ε)</span><br>
|
||||
<br> (<span class="emphasis"><em>GSL 1.16:</em></span> Max = 1.18e+03ε (Mean = 238ε))<br>
|
||||
(<span class="emphasis"><em>Rmath 3.0.2:</em></span> Max = 1.09e+03ε (Mean = 265ε))<br>
|
||||
(<span class="emphasis"><em>Cephes:</em></span> Max = 1.5e+03ε (Mean = 329ε))
|
||||
</p>
|
||||
</td>
|
||||
<td>
|
||||
<p>
|
||||
<span class="blue">Max = 107ε (Mean = 24.5ε)</span>
|
||||
</p>
|
||||
</td>
|
||||
</tr>
|
||||
<tr>
|
||||
<td>
|
||||
<p>
|
||||
Beta Function: Divergent Values
|
||||
</p>
|
||||
</td>
|
||||
<td>
|
||||
<p>
|
||||
<span class="blue">Max = 10.7ε (Mean = 2.22ε)</span>
|
||||
</p>
|
||||
</td>
|
||||
<td>
|
||||
<p>
|
||||
<span class="blue">Max = 10.3ε (Mean = 2.41ε)</span><br> <br>
|
||||
(<span class="emphasis"><em><tr1/cmath>:</em></span> Max = 128ε (Mean = 23.8ε))
|
||||
</p>
|
||||
</td>
|
||||
<td>
|
||||
<p>
|
||||
<span class="blue">Max = 0ε (Mean = 0ε)</span><br> <br> (<span class="emphasis"><em>GSL
|
||||
1.16:</em></span> Max = 12.1ε (Mean = 1.99ε))<br> (<span class="emphasis"><em>Rmath
|
||||
3.0.2:</em></span> Max = 176ε (Mean = 28ε))<br> (<span class="emphasis"><em>Cephes:</em></span>
|
||||
Max = 175ε (Mean = 22ε))
|
||||
</p>
|
||||
</td>
|
||||
<td>
|
||||
<p>
|
||||
<span class="blue">Max = 19.7ε (Mean = 2.75ε)</span>
|
||||
</p>
|
||||
</td>
|
||||
</tr>
|
||||
</tbody>
|
||||
</table></div>
|
||||
</div>
|
||||
<br class="table-break"><p>
|
||||
Note that the worst errors occur when a or b are large, and that when this
|
||||
is the case the result is very close to zero, so absolute errors will be
|
||||
very small.
|
||||
</p>
|
||||
<h5>
|
||||
<a name="math_toolkit.sf_beta.beta_function.h3"></a>
|
||||
<span class="phrase"><a name="math_toolkit.sf_beta.beta_function.testing"></a></span><a class="link" href="beta_function.html#math_toolkit.sf_beta.beta_function.testing">Testing</a>
|
||||
</h5>
|
||||
<p>
|
||||
A mixture of spot tests of exact values, and randomly generated test data
|
||||
are used: the test data was computed using <a href="http://shoup.net/ntl/doc/RR.txt" target="_top">NTL::RR</a>
|
||||
at 1000-bit precision.
|
||||
</p>
|
||||
<h5>
|
||||
<a name="math_toolkit.sf_beta.beta_function.h4"></a>
|
||||
<span class="phrase"><a name="math_toolkit.sf_beta.beta_function.implementation"></a></span><a class="link" href="beta_function.html#math_toolkit.sf_beta.beta_function.implementation">Implementation</a>
|
||||
</h5>
|
||||
<p>
|
||||
Traditional methods of evaluating the beta function either involve evaluating
|
||||
the gamma functions directly, or taking logarithms and then exponentiating
|
||||
the result. However, the former is prone to overflows for even very modest
|
||||
arguments, while the latter is prone to cancellation errors. As an alternative,
|
||||
if we regard the gamma function as a white-box containing the <a class="link" href="../lanczos.html" title="The Lanczos Approximation">Lanczos
|
||||
approximation</a>, then we can combine the power terms:
|
||||
</p>
|
||||
<p>
|
||||
<span class="inlinemediaobject"><img src="../../../equations/beta2.svg"></span>
|
||||
</p>
|
||||
<p>
|
||||
which is almost the ideal solution, however almost all of the error occurs
|
||||
in evaluating the power terms when <span class="emphasis"><em>a</em></span> or <span class="emphasis"><em>b</em></span>
|
||||
are large. If we assume that <span class="emphasis"><em>a > b</em></span> then the larger
|
||||
of the two power terms can be reduced by a factor of <span class="emphasis"><em>b</em></span>,
|
||||
which immediately cuts the maximum error in half:
|
||||
</p>
|
||||
<p>
|
||||
<span class="inlinemediaobject"><img src="../../../equations/beta3.svg"></span>
|
||||
</p>
|
||||
<p>
|
||||
This may not be the final solution, but it is very competitive compared to
|
||||
other implementation methods.
|
||||
</p>
|
||||
<p>
|
||||
The generic implementation - where no <a class="link" href="../lanczos.html" title="The Lanczos Approximation">Lanczos
|
||||
approximation</a> approximation is available - is implemented in a very
|
||||
similar way to the generic version of the gamma function. Again in order
|
||||
to avoid numerical overflow the power terms that prefix the series and continued
|
||||
fraction parts are collected together into:
|
||||
</p>
|
||||
<p>
|
||||
<span class="inlinemediaobject"><img src="../../../equations/beta8.svg"></span>
|
||||
</p>
|
||||
<p>
|
||||
where la, lb and lc are the integration limits used for a, b, and a+b.
|
||||
</p>
|
||||
<p>
|
||||
There are a few special cases worth mentioning:
|
||||
</p>
|
||||
<p>
|
||||
When <span class="emphasis"><em>a</em></span> or <span class="emphasis"><em>b</em></span> are less than one,
|
||||
we can use the recurrence relations:
|
||||
</p>
|
||||
<p>
|
||||
<span class="inlinemediaobject"><img src="../../../equations/beta4.svg"></span>
|
||||
</p>
|
||||
<p>
|
||||
<span class="inlinemediaobject"><img src="../../../equations/beta5.svg"></span>
|
||||
</p>
|
||||
<p>
|
||||
to move to a more favorable region where they are both greater than 1.
|
||||
</p>
|
||||
<p>
|
||||
In addition:
|
||||
</p>
|
||||
<p>
|
||||
<span class="inlinemediaobject"><img src="../../../equations/beta7.svg"></span>
|
||||
</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 © 2006-2010, 2012-2014 Nikhar Agrawal,
|
||||
Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos, Hubert
|
||||
Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Johan Råde, Gautam Sewani,
|
||||
Benjamin Sobotta, Thijs van den Berg, Daryle Walker and Xiaogang Zhang<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="../sf_beta.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_function.html"><img src="../../../../../../doc/src/images/next.png" alt="Next"></a>
|
||||
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|
||||
</body>
|
||||
</html>
|
||||
@@ -0,0 +1,893 @@
|
||||
<html>
|
||||
<head>
|
||||
<meta http-equiv="Content-Type" content="text/html; charset=US-ASCII">
|
||||
<title>Incomplete Beta Functions</title>
|
||||
<link rel="stylesheet" href="../../math.css" type="text/css">
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<meta name="generator" content="DocBook XSL Stylesheets V1.79.1">
|
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<link rel="home" href="../../index.html" title="Math Toolkit 2.5.1">
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<link rel="up" href="../sf_beta.html" title="Beta Functions">
|
||||
<link rel="prev" href="beta_function.html" title="Beta">
|
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<link rel="next" href="ibeta_inv_function.html" title="The Incomplete Beta Function Inverses">
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|
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<body bgcolor="white" text="black" link="#0000FF" vlink="#840084" alink="#0000FF">
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<table cellpadding="2" width="100%"><tr>
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<td align="center"><a href="../../../../../../index.html">Home</a></td>
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<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>
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</div>
|
||||
<div class="section">
|
||||
<div class="titlepage"><div><div><h3 class="title">
|
||||
<a name="math_toolkit.sf_beta.ibeta_function"></a><a class="link" href="ibeta_function.html" title="Incomplete Beta Functions">Incomplete Beta
|
||||
Functions</a>
|
||||
</h3></div></div></div>
|
||||
<h5>
|
||||
<a name="math_toolkit.sf_beta.ibeta_function.h0"></a>
|
||||
<span class="phrase"><a name="math_toolkit.sf_beta.ibeta_function.synopsis"></a></span><a class="link" href="ibeta_function.html#math_toolkit.sf_beta.ibeta_function.synopsis">Synopsis</a>
|
||||
</h5>
|
||||
<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>
|
||||
<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="../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="Chapter 15. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">></span>
|
||||
<a class="link" href="../result_type.html" title="Calculation of the Type of the Result"><span class="emphasis"><em>calculated-result-type</em></span></a> <span class="identifier">ibeta</span><span class="special">(</span><span class="identifier">T1</span> <span class="identifier">a</span><span class="special">,</span> <span class="identifier">T2</span> <span class="identifier">b</span><span class="special">,</span> <span class="identifier">T3</span> <span class="identifier">x</span><span class="special">,</span> <span class="keyword">const</span> <a class="link" href="../../policy.html" title="Chapter 15. Policies: Controlling Precision, Error Handling etc">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="../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="Chapter 15. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">></span>
|
||||
<a class="link" href="../result_type.html" title="Calculation of the Type of the Result"><span class="emphasis"><em>calculated-result-type</em></span></a> <span class="identifier">ibetac</span><span class="special">(</span><span class="identifier">T1</span> <span class="identifier">a</span><span class="special">,</span> <span class="identifier">T2</span> <span class="identifier">b</span><span class="special">,</span> <span class="identifier">T3</span> <span class="identifier">x</span><span class="special">,</span> <span class="keyword">const</span> <a class="link" href="../../policy.html" title="Chapter 15. Policies: Controlling Precision, Error Handling etc">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="../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="Chapter 15. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">></span>
|
||||
<a class="link" href="../result_type.html" title="Calculation of the Type of the Result"><span class="emphasis"><em>calculated-result-type</em></span></a> <span class="identifier">beta</span><span class="special">(</span><span class="identifier">T1</span> <span class="identifier">a</span><span class="special">,</span> <span class="identifier">T2</span> <span class="identifier">b</span><span class="special">,</span> <span class="identifier">T3</span> <span class="identifier">x</span><span class="special">,</span> <span class="keyword">const</span> <a class="link" href="../../policy.html" title="Chapter 15. Policies: Controlling Precision, Error Handling etc">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="../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="Chapter 15. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">></span>
|
||||
<a class="link" href="../result_type.html" title="Calculation of the Type of the Result"><span class="emphasis"><em>calculated-result-type</em></span></a> <span class="identifier">betac</span><span class="special">(</span><span class="identifier">T1</span> <span class="identifier">a</span><span class="special">,</span> <span class="identifier">T2</span> <span class="identifier">b</span><span class="special">,</span> <span class="identifier">T3</span> <span class="identifier">x</span><span class="special">,</span> <span class="keyword">const</span> <a class="link" href="../../policy.html" title="Chapter 15. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&);</span>
|
||||
|
||||
<span class="special">}}</span> <span class="comment">// namespaces</span>
|
||||
</pre>
|
||||
<h5>
|
||||
<a name="math_toolkit.sf_beta.ibeta_function.h1"></a>
|
||||
<span class="phrase"><a name="math_toolkit.sf_beta.ibeta_function.description"></a></span><a class="link" href="ibeta_function.html#math_toolkit.sf_beta.ibeta_function.description">Description</a>
|
||||
</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="../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>
|
||||
The final <a class="link" href="../../policy.html" title="Chapter 15. Policies: Controlling Precision, Error Handling etc">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="Chapter 15. Policies: Controlling Precision, Error Handling etc">policy
|
||||
documentation for more details</a>.
|
||||
</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="../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="Chapter 15. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">></span>
|
||||
<a class="link" href="../result_type.html" title="Calculation of the Type of the Result"><span class="emphasis"><em>calculated-result-type</em></span></a> <span class="identifier">ibeta</span><span class="special">(</span><span class="identifier">T1</span> <span class="identifier">a</span><span class="special">,</span> <span class="identifier">T2</span> <span class="identifier">b</span><span class="special">,</span> <span class="identifier">T3</span> <span class="identifier">x</span><span class="special">,</span> <span class="keyword">const</span> <a class="link" href="../../policy.html" title="Chapter 15. Policies: Controlling Precision, Error Handling etc">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.svg"></span>
|
||||
</p>
|
||||
<p>
|
||||
<span class="inlinemediaobject"><img src="../../../graphs/ibeta.svg" 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="../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="Chapter 15. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">></span>
|
||||
<a class="link" href="../result_type.html" title="Calculation of the Type of the Result"><span class="emphasis"><em>calculated-result-type</em></span></a> <span class="identifier">ibetac</span><span class="special">(</span><span class="identifier">T1</span> <span class="identifier">a</span><span class="special">,</span> <span class="identifier">T2</span> <span class="identifier">b</span><span class="special">,</span> <span class="identifier">T3</span> <span class="identifier">x</span><span class="special">,</span> <span class="keyword">const</span> <a class="link" href="../../policy.html" title="Chapter 15. Policies: Controlling Precision, Error Handling etc">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.svg"></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="../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="Chapter 15. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">></span>
|
||||
<a class="link" href="../result_type.html" title="Calculation of the Type of the Result"><span class="emphasis"><em>calculated-result-type</em></span></a> <span class="identifier">beta</span><span class="special">(</span><span class="identifier">T1</span> <span class="identifier">a</span><span class="special">,</span> <span class="identifier">T2</span> <span class="identifier">b</span><span class="special">,</span> <span class="identifier">T3</span> <span class="identifier">x</span><span class="special">,</span> <span class="keyword">const</span> <a class="link" href="../../policy.html" title="Chapter 15. Policies: Controlling Precision, Error Handling etc">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.svg"></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="../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="Chapter 15. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">></span>
|
||||
<a class="link" href="../result_type.html" title="Calculation of the Type of the Result"><span class="emphasis"><em>calculated-result-type</em></span></a> <span class="identifier">betac</span><span class="special">(</span><span class="identifier">T1</span> <span class="identifier">a</span><span class="special">,</span> <span class="identifier">T2</span> <span class="identifier">b</span><span class="special">,</span> <span class="identifier">T3</span> <span class="identifier">x</span><span class="special">,</span> <span class="keyword">const</span> <a class="link" href="../../policy.html" title="Chapter 15. Policies: Controlling Precision, Error Handling etc">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.svg"></span>
|
||||
</p>
|
||||
<h5>
|
||||
<a name="math_toolkit.sf_beta.ibeta_function.h2"></a>
|
||||
<span class="phrase"><a name="math_toolkit.sf_beta.ibeta_function.accuracy"></a></span><a class="link" href="ibeta_function.html#math_toolkit.sf_beta.ibeta_function.accuracy">Accuracy</a>
|
||||
</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="../relative_error.html#math_toolkit.relative_error.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.sf_beta.ibeta_function.table_ibeta"></a><p class="title"><b>Table 6.18. Error rates for ibeta</b></p>
|
||||
<div class="table-contents"><table class="table" summary="Error rates for ibeta">
|
||||
<colgroup>
|
||||
<col>
|
||||
<col>
|
||||
<col>
|
||||
<col>
|
||||
<col>
|
||||
</colgroup>
|
||||
<thead><tr>
|
||||
<th>
|
||||
</th>
|
||||
<th>
|
||||
<p>
|
||||
Microsoft Visual C++ version 12.0<br> Win32<br> double
|
||||
</p>
|
||||
</th>
|
||||
<th>
|
||||
<p>
|
||||
GNU C++ version 5.1.0<br> linux<br> double
|
||||
</p>
|
||||
</th>
|
||||
<th>
|
||||
<p>
|
||||
GNU C++ version 5.1.0<br> linux<br> long double
|
||||
</p>
|
||||
</th>
|
||||
<th>
|
||||
<p>
|
||||
Sun compiler version 0x5130<br> Sun Solaris<br> long double
|
||||
</p>
|
||||
</th>
|
||||
</tr></thead>
|
||||
<tbody>
|
||||
<tr>
|
||||
<td>
|
||||
<p>
|
||||
Incomplete Beta Function: Small Values
|
||||
</p>
|
||||
</td>
|
||||
<td>
|
||||
<p>
|
||||
<span class="blue">Max = 9.42ε (Mean = 2.24ε)</span>
|
||||
</p>
|
||||
</td>
|
||||
<td>
|
||||
<p>
|
||||
<span class="blue">Max = 0ε (Mean = 0ε)</span><br> <br> (<span class="emphasis"><em>GSL
|
||||
1.16:</em></span> Max = 682ε (Mean = 32.6ε))<br> (<span class="emphasis"><em>Rmath
|
||||
3.0.2:</em></span> Max = 22.9ε (Mean = 3.35ε))<br> (<span class="emphasis"><em>Cephes:</em></span>
|
||||
Max = 22.7ε (Mean = 1.69ε))
|
||||
</p>
|
||||
</td>
|
||||
<td>
|
||||
<p>
|
||||
<span class="blue">Max = 6.27ε (Mean = 1.7ε)</span>
|
||||
</p>
|
||||
</td>
|
||||
<td>
|
||||
<p>
|
||||
<span class="blue">Max = 21.3ε (Mean = 2.74ε)</span>
|
||||
</p>
|
||||
</td>
|
||||
</tr>
|
||||
<tr>
|
||||
<td>
|
||||
<p>
|
||||
Incomplete Beta Function: Medium Values
|
||||
</p>
|
||||
</td>
|
||||
<td>
|
||||
<p>
|
||||
<span class="blue">Max = 108ε (Mean = 16.3ε)</span>
|
||||
</p>
|
||||
</td>
|
||||
<td>
|
||||
<p>
|
||||
<span class="blue">Max = 0ε (Mean = 0ε)</span><br> <br> (<span class="emphasis"><em>GSL
|
||||
1.16:</em></span> Max = 690ε (Mean = 151ε))<br> (<span class="emphasis"><em>Rmath
|
||||
3.0.2:</em></span> Max = 232ε (Mean = 27.9ε))<br> (<span class="emphasis"><em>Cephes:</em></span>
|
||||
Max = 1.55e+03ε (Mean = 121ε))
|
||||
</p>
|
||||
</td>
|
||||
<td>
|
||||
<p>
|
||||
<span class="blue">Max = 51.4ε (Mean = 12ε)</span>
|
||||
</p>
|
||||
</td>
|
||||
<td>
|
||||
<p>
|
||||
<span class="blue">Max = 124ε (Mean = 18.4ε)</span>
|
||||
</p>
|
||||
</td>
|
||||
</tr>
|
||||
<tr>
|
||||
<td>
|
||||
<p>
|
||||
Incomplete Beta Function: Large and Diverse Values
|
||||
</p>
|
||||
</td>
|
||||
<td>
|
||||
<p>
|
||||
<span class="blue">Max = 1.19e+003ε (Mean = 59.5ε)</span>
|
||||
</p>
|
||||
</td>
|
||||
<td>
|
||||
<p>
|
||||
<span class="blue">Max = 1.26ε (Mean = 0.063ε)</span><br> <br>
|
||||
(<span class="emphasis"><em>GSL 1.16:</em></span> Max = 3.9e+05ε (Mean = 1.82e+04ε)
|
||||
<a class="link" href="../logs_and_tables/logs.html#errors_GNU_C_version_5_1_0_linux_double_ibeta_GSL_1_16_Incomplete_Beta_Function_Large_and_Diverse_Values">And
|
||||
other failures.</a>)<br> (<span class="emphasis"><em>Rmath 3.0.2:</em></span>
|
||||
Max = 574ε (Mean = 49.4ε))<br> (<span class="emphasis"><em>Cephes:</em></span> Max
|
||||
= 5.92e+05ε (Mean = 2.17e+04ε))
|
||||
</p>
|
||||
</td>
|
||||
<td>
|
||||
<p>
|
||||
<span class="blue">Max = 1.96e+04ε (Mean = 997ε)</span>
|
||||
</p>
|
||||
</td>
|
||||
<td>
|
||||
<p>
|
||||
<span class="blue">Max = 4.98e+04ε (Mean = 2.07e+03ε)</span>
|
||||
</p>
|
||||
</td>
|
||||
</tr>
|
||||
<tr>
|
||||
<td>
|
||||
<p>
|
||||
Incomplete Beta Function: Small Integer Values
|
||||
</p>
|
||||
</td>
|
||||
<td>
|
||||
<p>
|
||||
<span class="blue">Max = 3.69ε (Mean = 0.765ε)</span>
|
||||
</p>
|
||||
</td>
|
||||
<td>
|
||||
<p>
|
||||
<span class="blue">Max = 0ε (Mean = 0ε)</span><br> <br> (<span class="emphasis"><em>GSL
|
||||
1.16:</em></span> Max = 254ε (Mean = 50.9ε))<br> (<span class="emphasis"><em>Rmath
|
||||
3.0.2:</em></span> Max = 62.2ε (Mean = 8.95ε))<br> (<span class="emphasis"><em>Cephes:</em></span>
|
||||
Max = 7.96ε (Mean = 1.3ε))
|
||||
</p>
|
||||
</td>
|
||||
<td>
|
||||
<p>
|
||||
<span class="blue">Max = 4.45ε (Mean = 0.814ε)</span>
|
||||
</p>
|
||||
</td>
|
||||
<td>
|
||||
<p>
|
||||
<span class="blue">Max = 44.5ε (Mean = 10.1ε)</span>
|
||||
</p>
|
||||
</td>
|
||||
</tr>
|
||||
</tbody>
|
||||
</table></div>
|
||||
</div>
|
||||
<br class="table-break"><div class="table">
|
||||
<a name="math_toolkit.sf_beta.ibeta_function.table_ibetac"></a><p class="title"><b>Table 6.19. Error rates for ibetac</b></p>
|
||||
<div class="table-contents"><table class="table" summary="Error rates for ibetac">
|
||||
<colgroup>
|
||||
<col>
|
||||
<col>
|
||||
<col>
|
||||
<col>
|
||||
<col>
|
||||
</colgroup>
|
||||
<thead><tr>
|
||||
<th>
|
||||
</th>
|
||||
<th>
|
||||
<p>
|
||||
Microsoft Visual C++ version 12.0<br> Win32<br> double
|
||||
</p>
|
||||
</th>
|
||||
<th>
|
||||
<p>
|
||||
GNU C++ version 5.1.0<br> linux<br> double
|
||||
</p>
|
||||
</th>
|
||||
<th>
|
||||
<p>
|
||||
GNU C++ version 5.1.0<br> linux<br> long double
|
||||
</p>
|
||||
</th>
|
||||
<th>
|
||||
<p>
|
||||
Sun compiler version 0x5130<br> Sun Solaris<br> long double
|
||||
</p>
|
||||
</th>
|
||||
</tr></thead>
|
||||
<tbody>
|
||||
<tr>
|
||||
<td>
|
||||
<p>
|
||||
Incomplete Beta Function: Small Values
|
||||
</p>
|
||||
</td>
|
||||
<td>
|
||||
<p>
|
||||
<span class="blue">Max = 6.56ε (Mean = 1.88ε)</span>
|
||||
</p>
|
||||
</td>
|
||||
<td>
|
||||
<p>
|
||||
<span class="blue">Max = 0ε (Mean = 0ε)</span><br> <br> (<span class="emphasis"><em>Rmath
|
||||
3.0.2:</em></span> Max = 22.4ε (Mean = 3.67ε))
|
||||
</p>
|
||||
</td>
|
||||
<td>
|
||||
<p>
|
||||
<span class="blue">Max = 8.47ε (Mean = 1.68ε)</span>
|
||||
</p>
|
||||
</td>
|
||||
<td>
|
||||
<p>
|
||||
<span class="blue">Max = 13.8ε (Mean = 2.69ε)</span>
|
||||
</p>
|
||||
</td>
|
||||
</tr>
|
||||
<tr>
|
||||
<td>
|
||||
<p>
|
||||
Incomplete Beta Function: Medium Values
|
||||
</p>
|
||||
</td>
|
||||
<td>
|
||||
<p>
|
||||
<span class="blue">Max = 56.2ε (Mean = 14.1ε)</span>
|
||||
</p>
|
||||
</td>
|
||||
<td>
|
||||
<p>
|
||||
<span class="blue">Max = 0ε (Mean = 0ε)</span><br> <br> (<span class="emphasis"><em>Rmath
|
||||
3.0.2:</em></span> Max = 204ε (Mean = 25.8ε))
|
||||
</p>
|
||||
</td>
|
||||
<td>
|
||||
<p>
|
||||
<span class="blue">Max = 75.3ε (Mean = 11.3ε)</span>
|
||||
</p>
|
||||
</td>
|
||||
<td>
|
||||
<p>
|
||||
<span class="blue">Max = 132ε (Mean = 19.8ε)</span>
|
||||
</p>
|
||||
</td>
|
||||
</tr>
|
||||
<tr>
|
||||
<td>
|
||||
<p>
|
||||
Incomplete Beta Function: Large and Diverse Values
|
||||
</p>
|
||||
</td>
|
||||
<td>
|
||||
<p>
|
||||
<span class="blue">Max = 1.87e+003ε (Mean = 93.8ε)</span>
|
||||
</p>
|
||||
</td>
|
||||
<td>
|
||||
<p>
|
||||
<span class="blue">Max = 0.981ε (Mean = 0.0638ε)</span><br>
|
||||
<br> (<span class="emphasis"><em>Rmath 3.0.2:</em></span> Max = 889ε (Mean = 68.4ε))
|
||||
</p>
|
||||
</td>
|
||||
<td>
|
||||
<p>
|
||||
<span class="blue">Max = 3.45e+04ε (Mean = 1.32e+03ε)</span>
|
||||
</p>
|
||||
</td>
|
||||
<td>
|
||||
<p>
|
||||
<span class="blue">Max = 4.26e+04ε (Mean = 1.54e+03ε)</span>
|
||||
</p>
|
||||
</td>
|
||||
</tr>
|
||||
<tr>
|
||||
<td>
|
||||
<p>
|
||||
Incomplete Beta Function: Small Integer Values
|
||||
</p>
|
||||
</td>
|
||||
<td>
|
||||
<p>
|
||||
<span class="blue">Max = 6.37ε (Mean = 1.03ε)</span>
|
||||
</p>
|
||||
</td>
|
||||
<td>
|
||||
<p>
|
||||
<span class="blue">Max = 0ε (Mean = 0ε)</span><br> <br> (<span class="emphasis"><em>Rmath
|
||||
3.0.2:</em></span> Max = 84.6ε (Mean = 18ε))
|
||||
</p>
|
||||
</td>
|
||||
<td>
|
||||
<p>
|
||||
<span class="blue">Max = 5.34ε (Mean = 1.11ε)</span>
|
||||
</p>
|
||||
</td>
|
||||
<td>
|
||||
<p>
|
||||
<span class="blue">Max = 107ε (Mean = 17.1ε)</span>
|
||||
</p>
|
||||
</td>
|
||||
</tr>
|
||||
</tbody>
|
||||
</table></div>
|
||||
</div>
|
||||
<br class="table-break"><div class="table">
|
||||
<a name="math_toolkit.sf_beta.ibeta_function.table_beta_incomplete_"></a><p class="title"><b>Table 6.20. Error rates for beta (incomplete)</b></p>
|
||||
<div class="table-contents"><table class="table" summary="Error rates for beta (incomplete)">
|
||||
<colgroup>
|
||||
<col>
|
||||
<col>
|
||||
<col>
|
||||
<col>
|
||||
<col>
|
||||
</colgroup>
|
||||
<thead><tr>
|
||||
<th>
|
||||
</th>
|
||||
<th>
|
||||
<p>
|
||||
Microsoft Visual C++ version 12.0<br> Win32<br> double
|
||||
</p>
|
||||
</th>
|
||||
<th>
|
||||
<p>
|
||||
GNU C++ version 5.1.0<br> linux<br> double
|
||||
</p>
|
||||
</th>
|
||||
<th>
|
||||
<p>
|
||||
GNU C++ version 5.1.0<br> linux<br> long double
|
||||
</p>
|
||||
</th>
|
||||
<th>
|
||||
<p>
|
||||
Sun compiler version 0x5130<br> Sun Solaris<br> long double
|
||||
</p>
|
||||
</th>
|
||||
</tr></thead>
|
||||
<tbody>
|
||||
<tr>
|
||||
<td>
|
||||
<p>
|
||||
Incomplete Beta Function: Small Values
|
||||
</p>
|
||||
</td>
|
||||
<td>
|
||||
<p>
|
||||
<span class="blue">Max = 11.1ε (Mean = 2.28ε)</span>
|
||||
</p>
|
||||
</td>
|
||||
<td>
|
||||
<p>
|
||||
<span class="blue">Max = 0ε (Mean = 0ε)</span>
|
||||
</p>
|
||||
</td>
|
||||
<td>
|
||||
<p>
|
||||
<span class="blue">Max = 12.5ε (Mean = 2.01ε)</span>
|
||||
</p>
|
||||
</td>
|
||||
<td>
|
||||
<p>
|
||||
<span class="blue">Max = 18.7ε (Mean = 3.19ε)</span>
|
||||
</p>
|
||||
</td>
|
||||
</tr>
|
||||
<tr>
|
||||
<td>
|
||||
<p>
|
||||
Incomplete Beta Function: Medium Values
|
||||
</p>
|
||||
</td>
|
||||
<td>
|
||||
<p>
|
||||
<span class="blue">Max = 91.1ε (Mean = 12.7ε)</span>
|
||||
</p>
|
||||
</td>
|
||||
<td>
|
||||
<p>
|
||||
<span class="blue">Max = 0.568ε (Mean = 0.0342ε)</span>
|
||||
</p>
|
||||
</td>
|
||||
<td>
|
||||
<p>
|
||||
<span class="blue">Max = 67.1ε (Mean = 13.3ε)</span>
|
||||
</p>
|
||||
</td>
|
||||
<td>
|
||||
<p>
|
||||
<span class="blue">Max = 174ε (Mean = 25ε)</span>
|
||||
</p>
|
||||
</td>
|
||||
</tr>
|
||||
<tr>
|
||||
<td>
|
||||
<p>
|
||||
Incomplete Beta Function: Large and Diverse Values
|
||||
</p>
|
||||
</td>
|
||||
<td>
|
||||
<p>
|
||||
<span class="blue">Max = 635ε (Mean = 29.5ε)</span>
|
||||
</p>
|
||||
</td>
|
||||
<td>
|
||||
<p>
|
||||
<span class="blue">Max = 0.999ε (Mean = 0.0399ε)</span>
|
||||
</p>
|
||||
</td>
|
||||
<td>
|
||||
<p>
|
||||
<span class="blue">Max = 6.84e+04ε (Mean = 2.76e+03ε)</span>
|
||||
</p>
|
||||
</td>
|
||||
<td>
|
||||
<p>
|
||||
<span class="blue">Max = 6.86e+04ε (Mean = 2.79e+03ε)</span>
|
||||
</p>
|
||||
</td>
|
||||
</tr>
|
||||
<tr>
|
||||
<td>
|
||||
<p>
|
||||
Incomplete Beta Function: Small Integer Values
|
||||
</p>
|
||||
</td>
|
||||
<td>
|
||||
<p>
|
||||
<span class="blue">Max = 26.8ε (Mean = 6.61ε)</span>
|
||||
</p>
|
||||
</td>
|
||||
<td>
|
||||
<p>
|
||||
<span class="blue">Max = 0.786ε (Mean = 0.0248ε)</span>
|
||||
</p>
|
||||
</td>
|
||||
<td>
|
||||
<p>
|
||||
<span class="blue">Max = 10.2ε (Mean = 3.49ε)</span>
|
||||
</p>
|
||||
</td>
|
||||
<td>
|
||||
<p>
|
||||
<span class="blue">Max = 51.8ε (Mean = 11ε)</span>
|
||||
</p>
|
||||
</td>
|
||||
</tr>
|
||||
</tbody>
|
||||
</table></div>
|
||||
</div>
|
||||
<br class="table-break"><div class="table">
|
||||
<a name="math_toolkit.sf_beta.ibeta_function.table_betac"></a><p class="title"><b>Table 6.21. Error rates for betac</b></p>
|
||||
<div class="table-contents"><table class="table" summary="Error rates for betac">
|
||||
<colgroup>
|
||||
<col>
|
||||
<col>
|
||||
<col>
|
||||
<col>
|
||||
<col>
|
||||
</colgroup>
|
||||
<thead><tr>
|
||||
<th>
|
||||
</th>
|
||||
<th>
|
||||
<p>
|
||||
Microsoft Visual C++ version 12.0<br> Win32<br> double
|
||||
</p>
|
||||
</th>
|
||||
<th>
|
||||
<p>
|
||||
GNU C++ version 5.1.0<br> linux<br> double
|
||||
</p>
|
||||
</th>
|
||||
<th>
|
||||
<p>
|
||||
GNU C++ version 5.1.0<br> linux<br> long double
|
||||
</p>
|
||||
</th>
|
||||
<th>
|
||||
<p>
|
||||
Sun compiler version 0x5130<br> Sun Solaris<br> long double
|
||||
</p>
|
||||
</th>
|
||||
</tr></thead>
|
||||
<tbody>
|
||||
<tr>
|
||||
<td>
|
||||
<p>
|
||||
Incomplete Beta Function: Small Values
|
||||
</p>
|
||||
</td>
|
||||
<td>
|
||||
<p>
|
||||
<span class="blue">Max = 12ε (Mean = 2.43ε)</span>
|
||||
</p>
|
||||
</td>
|
||||
<td>
|
||||
<p>
|
||||
<span class="blue">Max = 0.676ε (Mean = 0.0302ε)</span>
|
||||
</p>
|
||||
</td>
|
||||
<td>
|
||||
<p>
|
||||
<span class="blue">Max = 7.75ε (Mean = 1.91ε)</span>
|
||||
</p>
|
||||
</td>
|
||||
<td>
|
||||
<p>
|
||||
<span class="blue">Max = 12.6ε (Mean = 3.02ε)</span>
|
||||
</p>
|
||||
</td>
|
||||
</tr>
|
||||
<tr>
|
||||
<td>
|
||||
<p>
|
||||
Incomplete Beta Function: Medium Values
|
||||
</p>
|
||||
</td>
|
||||
<td>
|
||||
<p>
|
||||
<span class="blue">Max = 91.3ε (Mean = 14.8ε)</span>
|
||||
</p>
|
||||
</td>
|
||||
<td>
|
||||
<p>
|
||||
<span class="blue">Max = 0.949ε (Mean = 0.0944ε)</span>
|
||||
</p>
|
||||
</td>
|
||||
<td>
|
||||
<p>
|
||||
<span class="blue">Max = 63.5ε (Mean = 13.4ε)</span>
|
||||
</p>
|
||||
</td>
|
||||
<td>
|
||||
<p>
|
||||
<span class="blue">Max = 97.6ε (Mean = 24.3ε)</span>
|
||||
</p>
|
||||
</td>
|
||||
</tr>
|
||||
<tr>
|
||||
<td>
|
||||
<p>
|
||||
Incomplete Beta Function: Large and Diverse Values
|
||||
</p>
|
||||
</td>
|
||||
<td>
|
||||
<p>
|
||||
<span class="blue">Max = 3.72e+003ε (Mean = 113ε)</span>
|
||||
</p>
|
||||
</td>
|
||||
<td>
|
||||
<p>
|
||||
<span class="blue">Max = 1.12ε (Mean = 0.0458ε)</span>
|
||||
</p>
|
||||
</td>
|
||||
<td>
|
||||
<p>
|
||||
<span class="blue">Max = 1.05e+05ε (Mean = 5.45e+03ε)</span>
|
||||
</p>
|
||||
</td>
|
||||
<td>
|
||||
<p>
|
||||
<span class="blue">Max = 1.04e+05ε (Mean = 5.46e+03ε)</span>
|
||||
</p>
|
||||
</td>
|
||||
</tr>
|
||||
<tr>
|
||||
<td>
|
||||
<p>
|
||||
Incomplete Beta Function: Small Integer Values
|
||||
</p>
|
||||
</td>
|
||||
<td>
|
||||
<p>
|
||||
<span class="blue">Max = 26.7ε (Mean = 6.67ε)</span>
|
||||
</p>
|
||||
</td>
|
||||
<td>
|
||||
<p>
|
||||
<span class="blue">Max = 0.586ε (Mean = 0.0314ε)</span>
|
||||
</p>
|
||||
</td>
|
||||
<td>
|
||||
<p>
|
||||
<span class="blue">Max = 10.4ε (Mean = 3.54ε)</span>
|
||||
</p>
|
||||
</td>
|
||||
<td>
|
||||
<p>
|
||||
<span class="blue">Max = 103ε (Mean = 17.4ε)</span>
|
||||
</p>
|
||||
</td>
|
||||
</tr>
|
||||
</tbody>
|
||||
</table></div>
|
||||
</div>
|
||||
<br class="table-break"><h5>
|
||||
<a name="math_toolkit.sf_beta.ibeta_function.h3"></a>
|
||||
<span class="phrase"><a name="math_toolkit.sf_beta.ibeta_function.testing"></a></span><a class="link" href="ibeta_function.html#math_toolkit.sf_beta.ibeta_function.testing">Testing</a>
|
||||
</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>
|
||||
<h5>
|
||||
<a name="math_toolkit.sf_beta.ibeta_function.h4"></a>
|
||||
<span class="phrase"><a name="math_toolkit.sf_beta.ibeta_function.implementation"></a></span><a class="link" href="ibeta_function.html#math_toolkit.sf_beta.ibeta_function.implementation">Implementation</a>
|
||||
</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.svg"></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.svg"></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.svg"></span>
|
||||
</p>
|
||||
<p>
|
||||
For smallish b and x then a series representation can be used:
|
||||
</p>
|
||||
<p>
|
||||
<span class="inlinemediaobject"><img src="../../../equations/ibeta7.svg"></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.svg"></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.svg"></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.svg"></span>
|
||||
</p>
|
||||
<p>
|
||||
<span class="inlinemediaobject"><img src="../../../equations/ibeta11.svg"></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 © 2006-2010, 2012-2014 Nikhar Agrawal,
|
||||
Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos, Hubert
|
||||
Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Johan Råde, Gautam Sewani,
|
||||
Benjamin Sobotta, Thijs van den Berg, Daryle Walker and Xiaogang Zhang<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>
|
||||
@@ -0,0 +1,912 @@
|
||||
<html>
|
||||
<head>
|
||||
<meta http-equiv="Content-Type" content="text/html; charset=US-ASCII">
|
||||
<title>The Incomplete Beta Function Inverses</title>
|
||||
<link rel="stylesheet" href="../../math.css" type="text/css">
|
||||
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|
||||
<link rel="home" href="../../index.html" title="Math Toolkit 2.5.1">
|
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<link rel="up" href="../sf_beta.html" title="Beta Functions">
|
||||
<link rel="prev" href="ibeta_function.html" title="Incomplete Beta Functions">
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<link rel="next" href="beta_derivative.html" title="Derivative of the Incomplete Beta Function">
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<body bgcolor="white" text="black" link="#0000FF" vlink="#840084" alink="#0000FF">
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<table cellpadding="2" width="100%"><tr>
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<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>
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<div class="section">
|
||||
<div class="titlepage"><div><div><h3 class="title">
|
||||
<a name="math_toolkit.sf_beta.ibeta_inv_function"></a><a class="link" href="ibeta_inv_function.html" title="The Incomplete Beta Function Inverses">The Incomplete
|
||||
Beta Function Inverses</a>
|
||||
</h3></div></div></div>
|
||||
<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>
|
||||
<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="../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="Chapter 15. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">></span>
|
||||
<a class="link" href="../result_type.html" title="Calculation of the Type of the Result"><span class="emphasis"><em>calculated-result-type</em></span></a> <span class="identifier">ibeta_inv</span><span class="special">(</span><span class="identifier">T1</span> <span class="identifier">a</span><span class="special">,</span> <span class="identifier">T2</span> <span class="identifier">b</span><span class="special">,</span> <span class="identifier">T3</span> <span class="identifier">p</span><span class="special">,</span> <span class="keyword">const</span> <a class="link" href="../../policy.html" title="Chapter 15. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&);</span>
|
||||
|
||||
<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="../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="Chapter 15. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">></span>
|
||||
<a class="link" href="../result_type.html" title="Calculation of the Type of the Result"><span class="emphasis"><em>calculated-result-type</em></span></a> <span class="identifier">ibeta_inv</span><span class="special">(</span><span class="identifier">T1</span> <span class="identifier">a</span><span class="special">,</span> <span class="identifier">T2</span> <span class="identifier">b</span><span class="special">,</span> <span class="identifier">T3</span> <span class="identifier">p</span><span class="special">,</span> <span class="identifier">T4</span><span class="special">*</span> <span class="identifier">py</span><span class="special">,</span> <span class="keyword">const</span> <a class="link" href="../../policy.html" title="Chapter 15. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&);</span>
|
||||
|
||||
<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="../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="Chapter 15. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">></span>
|
||||
<a class="link" href="../result_type.html" title="Calculation of the Type of the Result"><span class="emphasis"><em>calculated-result-type</em></span></a> <span class="identifier">ibetac_inv</span><span class="special">(</span><span class="identifier">T1</span> <span class="identifier">a</span><span class="special">,</span> <span class="identifier">T2</span> <span class="identifier">b</span><span class="special">,</span> <span class="identifier">T3</span> <span class="identifier">q</span><span class="special">,</span> <span class="keyword">const</span> <a class="link" href="../../policy.html" title="Chapter 15. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&);</span>
|
||||
|
||||
<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="../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="Chapter 15. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">></span>
|
||||
<a class="link" href="../result_type.html" title="Calculation of the Type of the Result"><span class="emphasis"><em>calculated-result-type</em></span></a> <span class="identifier">ibetac_inv</span><span class="special">(</span><span class="identifier">T1</span> <span class="identifier">a</span><span class="special">,</span> <span class="identifier">T2</span> <span class="identifier">b</span><span class="special">,</span> <span class="identifier">T3</span> <span class="identifier">q</span><span class="special">,</span> <span class="identifier">T4</span><span class="special">*</span> <span class="identifier">py</span><span class="special">,</span> <span class="keyword">const</span> <a class="link" href="../../policy.html" title="Chapter 15. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&);</span>
|
||||
|
||||
<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="../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="Chapter 15. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">></span>
|
||||
<a class="link" href="../result_type.html" title="Calculation of the Type of the Result"><span class="emphasis"><em>calculated-result-type</em></span></a> <span class="identifier">ibeta_inva</span><span class="special">(</span><span class="identifier">T1</span> <span class="identifier">b</span><span class="special">,</span> <span class="identifier">T2</span> <span class="identifier">x</span><span class="special">,</span> <span class="identifier">T3</span> <span class="identifier">p</span><span class="special">,</span> <span class="keyword">const</span> <a class="link" href="../../policy.html" title="Chapter 15. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&);</span>
|
||||
|
||||
<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="../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="Chapter 15. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">></span>
|
||||
<a class="link" href="../result_type.html" title="Calculation of the Type of the Result"><span class="emphasis"><em>calculated-result-type</em></span></a> <span class="identifier">ibetac_inva</span><span class="special">(</span><span class="identifier">T1</span> <span class="identifier">b</span><span class="special">,</span> <span class="identifier">T2</span> <span class="identifier">x</span><span class="special">,</span> <span class="identifier">T3</span> <span class="identifier">q</span><span class="special">,</span> <span class="keyword">const</span> <a class="link" href="../../policy.html" title="Chapter 15. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&);</span>
|
||||
|
||||
<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="../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="Chapter 15. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">></span>
|
||||
<a class="link" href="../result_type.html" title="Calculation of the Type of the Result"><span class="emphasis"><em>calculated-result-type</em></span></a> <span class="identifier">ibeta_invb</span><span class="special">(</span><span class="identifier">T1</span> <span class="identifier">a</span><span class="special">,</span> <span class="identifier">T2</span> <span class="identifier">x</span><span class="special">,</span> <span class="identifier">T3</span> <span class="identifier">p</span><span class="special">,</span> <span class="keyword">const</span> <a class="link" href="../../policy.html" title="Chapter 15. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&);</span>
|
||||
|
||||
<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="../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="Chapter 15. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">></span>
|
||||
<a class="link" href="../result_type.html" title="Calculation of the Type of the Result"><span class="emphasis"><em>calculated-result-type</em></span></a> <span class="identifier">ibetac_invb</span><span class="special">(</span><span class="identifier">T1</span> <span class="identifier">a</span><span class="special">,</span> <span class="identifier">T2</span> <span class="identifier">x</span><span class="special">,</span> <span class="identifier">T3</span> <span class="identifier">q</span><span class="special">,</span> <span class="keyword">const</span> <a class="link" href="../../policy.html" title="Chapter 15. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&);</span>
|
||||
|
||||
<span class="special">}}</span> <span class="comment">// namespaces</span>
|
||||
</pre>
|
||||
<h5>
|
||||
<a name="math_toolkit.sf_beta.ibeta_inv_function.h0"></a>
|
||||
<span class="phrase"><a name="math_toolkit.sf_beta.ibeta_inv_function.description"></a></span><a class="link" href="ibeta_inv_function.html#math_toolkit.sf_beta.ibeta_inv_function.description">Description</a>
|
||||
</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>
|
||||
The final <a class="link" href="../../policy.html" title="Chapter 15. Policies: Controlling Precision, Error Handling etc">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="Chapter 15. Policies: Controlling Precision, Error Handling etc">policy
|
||||
documentation for more details</a>.
|
||||
</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 ibetac_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="../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="../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="Chapter 15. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">></span>
|
||||
<a class="link" href="../result_type.html" title="Calculation of the Type of the Result"><span class="emphasis"><em>calculated-result-type</em></span></a> <span class="identifier">ibeta_inv</span><span class="special">(</span><span class="identifier">T1</span> <span class="identifier">a</span><span class="special">,</span> <span class="identifier">T2</span> <span class="identifier">b</span><span class="special">,</span> <span class="identifier">T3</span> <span class="identifier">p</span><span class="special">,</span> <span class="keyword">const</span> <a class="link" href="../../policy.html" title="Chapter 15. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&);</span>
|
||||
|
||||
<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="../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="Chapter 15. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">></span>
|
||||
<a class="link" href="../result_type.html" title="Calculation of the Type of the Result"><span class="emphasis"><em>calculated-result-type</em></span></a> <span class="identifier">ibeta_inv</span><span class="special">(</span><span class="identifier">T1</span> <span class="identifier">a</span><span class="special">,</span> <span class="identifier">T2</span> <span class="identifier">b</span><span class="special">,</span> <span class="identifier">T3</span> <span class="identifier">p</span><span class="special">,</span> <span class="identifier">T4</span><span class="special">*</span> <span class="identifier">py</span><span class="special">,</span> <span class="keyword">const</span> <a class="link" href="../../policy.html" title="Chapter 15. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&);</span>
|
||||
</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>
|
||||
The final <a class="link" href="../../policy.html" title="Chapter 15. Policies: Controlling Precision, Error Handling etc">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="Chapter 15. Policies: Controlling Precision, Error Handling etc">policy
|
||||
documentation for more details</a>.
|
||||
</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="../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="Chapter 15. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">></span>
|
||||
<a class="link" href="../result_type.html" title="Calculation of the Type of the Result"><span class="emphasis"><em>calculated-result-type</em></span></a> <span class="identifier">ibetac_inv</span><span class="special">(</span><span class="identifier">T1</span> <span class="identifier">a</span><span class="special">,</span> <span class="identifier">T2</span> <span class="identifier">b</span><span class="special">,</span> <span class="identifier">T3</span> <span class="identifier">q</span><span class="special">,</span> <span class="keyword">const</span> <a class="link" href="../../policy.html" title="Chapter 15. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&);</span>
|
||||
|
||||
<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="../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="Chapter 15. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">></span>
|
||||
<a class="link" href="../result_type.html" title="Calculation of the Type of the Result"><span class="emphasis"><em>calculated-result-type</em></span></a> <span class="identifier">ibetac_inv</span><span class="special">(</span><span class="identifier">T1</span> <span class="identifier">a</span><span class="special">,</span> <span class="identifier">T2</span> <span class="identifier">b</span><span class="special">,</span> <span class="identifier">T3</span> <span class="identifier">q</span><span class="special">,</span> <span class="identifier">T4</span><span class="special">*</span> <span class="identifier">py</span><span class="special">,</span> <span class="keyword">const</span> <a class="link" href="../../policy.html" title="Chapter 15. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&);</span>
|
||||
</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>
|
||||
The final <a class="link" href="../../policy.html" title="Chapter 15. Policies: Controlling Precision, Error Handling etc">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="Chapter 15. Policies: Controlling Precision, Error Handling etc">policy
|
||||
documentation for more details</a>.
|
||||
</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="../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="Chapter 15. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">></span>
|
||||
<a class="link" href="../result_type.html" title="Calculation of the Type of the Result"><span class="emphasis"><em>calculated-result-type</em></span></a> <span class="identifier">ibeta_inva</span><span class="special">(</span><span class="identifier">T1</span> <span class="identifier">b</span><span class="special">,</span> <span class="identifier">T2</span> <span class="identifier">x</span><span class="special">,</span> <span class="identifier">T3</span> <span class="identifier">p</span><span class="special">,</span> <span class="keyword">const</span> <a class="link" href="../../policy.html" title="Chapter 15. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&);</span>
|
||||
</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>
|
||||
The final <a class="link" href="../../policy.html" title="Chapter 15. Policies: Controlling Precision, Error Handling etc">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="Chapter 15. Policies: Controlling Precision, Error Handling etc">policy
|
||||
documentation for more details</a>.
|
||||
</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="../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="Chapter 15. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">></span>
|
||||
<a class="link" href="../result_type.html" title="Calculation of the Type of the Result"><span class="emphasis"><em>calculated-result-type</em></span></a> <span class="identifier">ibetac_inva</span><span class="special">(</span><span class="identifier">T1</span> <span class="identifier">b</span><span class="special">,</span> <span class="identifier">T2</span> <span class="identifier">x</span><span class="special">,</span> <span class="identifier">T3</span> <span class="identifier">p</span><span class="special">,</span> <span class="keyword">const</span> <a class="link" href="../../policy.html" title="Chapter 15. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&);</span>
|
||||
</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>
|
||||
The final <a class="link" href="../../policy.html" title="Chapter 15. Policies: Controlling Precision, Error Handling etc">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="Chapter 15. Policies: Controlling Precision, Error Handling etc">policy
|
||||
documentation for more details</a>.
|
||||
</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="../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="Chapter 15. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">></span>
|
||||
<a class="link" href="../result_type.html" title="Calculation of the Type of the Result"><span class="emphasis"><em>calculated-result-type</em></span></a> <span class="identifier">ibeta_invb</span><span class="special">(</span><span class="identifier">T1</span> <span class="identifier">b</span><span class="special">,</span> <span class="identifier">T2</span> <span class="identifier">x</span><span class="special">,</span> <span class="identifier">T3</span> <span class="identifier">p</span><span class="special">,</span> <span class="keyword">const</span> <a class="link" href="../../policy.html" title="Chapter 15. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&);</span>
|
||||
</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>
|
||||
The final <a class="link" href="../../policy.html" title="Chapter 15. Policies: Controlling Precision, Error Handling etc">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="Chapter 15. Policies: Controlling Precision, Error Handling etc">policy
|
||||
documentation for more details</a>.
|
||||
</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="../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="Chapter 15. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">></span>
|
||||
<a class="link" href="../result_type.html" title="Calculation of the Type of the Result"><span class="emphasis"><em>calculated-result-type</em></span></a> <span class="identifier">ibetac_invb</span><span class="special">(</span><span class="identifier">T1</span> <span class="identifier">b</span><span class="special">,</span> <span class="identifier">T2</span> <span class="identifier">x</span><span class="special">,</span> <span class="identifier">T3</span> <span class="identifier">p</span><span class="special">,</span> <span class="keyword">const</span> <a class="link" href="../../policy.html" title="Chapter 15. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&);</span>
|
||||
</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>
|
||||
The final <a class="link" href="../../policy.html" title="Chapter 15. Policies: Controlling Precision, Error Handling etc">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="Chapter 15. Policies: Controlling Precision, Error Handling etc">policy
|
||||
documentation for more details</a>.
|
||||
</p>
|
||||
<h5>
|
||||
<a name="math_toolkit.sf_beta.ibeta_inv_function.h1"></a>
|
||||
<span class="phrase"><a name="math_toolkit.sf_beta.ibeta_inv_function.accuracy"></a></span><a class="link" href="ibeta_inv_function.html#math_toolkit.sf_beta.ibeta_inv_function.accuracy">Accuracy</a>
|
||||
</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>
|
||||
<p>
|
||||
Comparisons to other libraries are shown below, note that our test data exercises
|
||||
some rather extreme cases in the incomplete beta function which many other
|
||||
libraries fail to handle:
|
||||
</p>
|
||||
<div class="table">
|
||||
<a name="math_toolkit.sf_beta.ibeta_inv_function.table_ibeta_inv"></a><p class="title"><b>Table 6.22. Error rates for ibeta_inv</b></p>
|
||||
<div class="table-contents"><table class="table" summary="Error rates for ibeta_inv">
|
||||
<colgroup>
|
||||
<col>
|
||||
<col>
|
||||
<col>
|
||||
<col>
|
||||
<col>
|
||||
</colgroup>
|
||||
<thead><tr>
|
||||
<th>
|
||||
</th>
|
||||
<th>
|
||||
<p>
|
||||
Microsoft Visual C++ version 12.0<br> Win32<br> double
|
||||
</p>
|
||||
</th>
|
||||
<th>
|
||||
<p>
|
||||
GNU C++ version 5.1.0<br> linux<br> double
|
||||
</p>
|
||||
</th>
|
||||
<th>
|
||||
<p>
|
||||
GNU C++ version 5.1.0<br> linux<br> long double
|
||||
</p>
|
||||
</th>
|
||||
<th>
|
||||
<p>
|
||||
Sun compiler version 0x5130<br> Sun Solaris<br> long double
|
||||
</p>
|
||||
</th>
|
||||
</tr></thead>
|
||||
<tbody><tr>
|
||||
<td>
|
||||
<p>
|
||||
Inverse incomplete beta
|
||||
</p>
|
||||
</td>
|
||||
<td>
|
||||
<p>
|
||||
<span class="blue">Max = 7.08e+003ε (Mean = 244ε)</span>
|
||||
</p>
|
||||
</td>
|
||||
<td>
|
||||
<p>
|
||||
<span class="blue">Max = 3.21ε (Mean = 0.158ε)</span><br> <br>
|
||||
(<span class="emphasis"><em>Rmath 3.0.2:</em></span> <span class="red">Max = +INFε (Mean
|
||||
= +INFε) <a class="link" href="../logs_and_tables/logs.html#errors_GNU_C_version_5_1_0_linux_double_ibeta_inv_Rmath_3_0_2_Inverse_incomplete_beta">And
|
||||
other failures.</a>)</span><br> (<span class="emphasis"><em>Cephes:</em></span>
|
||||
<span class="red">Max = 1.28e+10ε (Mean = 5.17e+08ε))</span>
|
||||
</p>
|
||||
</td>
|
||||
<td>
|
||||
<p>
|
||||
<span class="blue">Max = 4.53e+04ε (Mean = 2.93e+03ε)</span>
|
||||
</p>
|
||||
</td>
|
||||
<td>
|
||||
<p>
|
||||
<span class="blue">Max = 4.22e+04ε (Mean = 2.8e+03ε)</span>
|
||||
</p>
|
||||
</td>
|
||||
</tr></tbody>
|
||||
</table></div>
|
||||
</div>
|
||||
<br class="table-break"><div class="table">
|
||||
<a name="math_toolkit.sf_beta.ibeta_inv_function.table_ibetac_inv"></a><p class="title"><b>Table 6.23. Error rates for ibetac_inv</b></p>
|
||||
<div class="table-contents"><table class="table" summary="Error rates for ibetac_inv">
|
||||
<colgroup>
|
||||
<col>
|
||||
<col>
|
||||
<col>
|
||||
<col>
|
||||
<col>
|
||||
</colgroup>
|
||||
<thead><tr>
|
||||
<th>
|
||||
</th>
|
||||
<th>
|
||||
<p>
|
||||
Microsoft Visual C++ version 12.0<br> Win32<br> double
|
||||
</p>
|
||||
</th>
|
||||
<th>
|
||||
<p>
|
||||
GNU C++ version 5.1.0<br> linux<br> double
|
||||
</p>
|
||||
</th>
|
||||
<th>
|
||||
<p>
|
||||
GNU C++ version 5.1.0<br> linux<br> long double
|
||||
</p>
|
||||
</th>
|
||||
<th>
|
||||
<p>
|
||||
Sun compiler version 0x5130<br> Sun Solaris<br> long double
|
||||
</p>
|
||||
</th>
|
||||
</tr></thead>
|
||||
<tbody><tr>
|
||||
<td>
|
||||
<p>
|
||||
Inverse incomplete beta
|
||||
</p>
|
||||
</td>
|
||||
<td>
|
||||
<p>
|
||||
<span class="blue">Max = 5.53e+003ε (Mean = 220ε)</span>
|
||||
</p>
|
||||
</td>
|
||||
<td>
|
||||
<p>
|
||||
<span class="blue">Max = 1.71ε (Mean = 0.108ε)</span><br> <br>
|
||||
(<span class="emphasis"><em>Rmath 3.0.2:</em></span> <span class="red">Max = +INFε (Mean
|
||||
= +INFε) <a class="link" href="../logs_and_tables/logs.html#errors_GNU_C_version_5_1_0_linux_double_ibetac_inv_Rmath_3_0_2_Inverse_incomplete_beta">And
|
||||
other failures.</a>)</span><br> (<span class="emphasis"><em>Cephes:</em></span>
|
||||
Max = 1.71ε (Mean = 0.108ε))
|
||||
</p>
|
||||
</td>
|
||||
<td>
|
||||
<p>
|
||||
<span class="blue">Max = 6.17e+04ε (Mean = 3.77e+03ε)</span>
|
||||
</p>
|
||||
</td>
|
||||
<td>
|
||||
<p>
|
||||
<span class="blue">Max = 5.15e+04ε (Mean = 3.51e+03ε)</span>
|
||||
</p>
|
||||
</td>
|
||||
</tr></tbody>
|
||||
</table></div>
|
||||
</div>
|
||||
<br class="table-break"><div class="table">
|
||||
<a name="math_toolkit.sf_beta.ibeta_inv_function.table_ibeta_inva"></a><p class="title"><b>Table 6.24. Error rates for ibeta_inva</b></p>
|
||||
<div class="table-contents"><table class="table" summary="Error rates for ibeta_inva">
|
||||
<colgroup>
|
||||
<col>
|
||||
<col>
|
||||
<col>
|
||||
<col>
|
||||
<col>
|
||||
</colgroup>
|
||||
<thead><tr>
|
||||
<th>
|
||||
</th>
|
||||
<th>
|
||||
<p>
|
||||
Microsoft Visual C++ version 12.0<br> Win32<br> double
|
||||
</p>
|
||||
</th>
|
||||
<th>
|
||||
<p>
|
||||
GNU C++ version 5.1.0<br> linux<br> double
|
||||
</p>
|
||||
</th>
|
||||
<th>
|
||||
<p>
|
||||
GNU C++ version 5.1.0<br> linux<br> long double
|
||||
</p>
|
||||
</th>
|
||||
<th>
|
||||
<p>
|
||||
Sun compiler version 0x5130<br> Sun Solaris<br> long double
|
||||
</p>
|
||||
</th>
|
||||
</tr></thead>
|
||||
<tbody><tr>
|
||||
<td>
|
||||
<p>
|
||||
Inverse incomplete beta
|
||||
</p>
|
||||
</td>
|
||||
<td>
|
||||
<p>
|
||||
<span class="blue">Max = 395ε (Mean = 24.7ε)</span>
|
||||
</p>
|
||||
</td>
|
||||
<td>
|
||||
<p>
|
||||
<span class="blue">Max = 0.602ε (Mean = 0.0239ε)</span>
|
||||
</p>
|
||||
</td>
|
||||
<td>
|
||||
<p>
|
||||
<span class="blue">Max = 377ε (Mean = 25.1ε)</span>
|
||||
</p>
|
||||
</td>
|
||||
<td>
|
||||
<p>
|
||||
<span class="blue">Max = 438ε (Mean = 31.6ε)</span>
|
||||
</p>
|
||||
</td>
|
||||
</tr></tbody>
|
||||
</table></div>
|
||||
</div>
|
||||
<br class="table-break"><div class="table">
|
||||
<a name="math_toolkit.sf_beta.ibeta_inv_function.table_ibetac_inva"></a><p class="title"><b>Table 6.25. Error rates for ibetac_inva</b></p>
|
||||
<div class="table-contents"><table class="table" summary="Error rates for ibetac_inva">
|
||||
<colgroup>
|
||||
<col>
|
||||
<col>
|
||||
<col>
|
||||
<col>
|
||||
<col>
|
||||
</colgroup>
|
||||
<thead><tr>
|
||||
<th>
|
||||
</th>
|
||||
<th>
|
||||
<p>
|
||||
Microsoft Visual C++ version 12.0<br> Win32<br> double
|
||||
</p>
|
||||
</th>
|
||||
<th>
|
||||
<p>
|
||||
GNU C++ version 5.1.0<br> linux<br> double
|
||||
</p>
|
||||
</th>
|
||||
<th>
|
||||
<p>
|
||||
GNU C++ version 5.1.0<br> linux<br> long double
|
||||
</p>
|
||||
</th>
|
||||
<th>
|
||||
<p>
|
||||
Sun compiler version 0x5130<br> Sun Solaris<br> long double
|
||||
</p>
|
||||
</th>
|
||||
</tr></thead>
|
||||
<tbody><tr>
|
||||
<td>
|
||||
<p>
|
||||
Inverse incomplete beta
|
||||
</p>
|
||||
</td>
|
||||
<td>
|
||||
<p>
|
||||
<span class="blue">Max = 408ε (Mean = 27.8ε)</span>
|
||||
</p>
|
||||
</td>
|
||||
<td>
|
||||
<p>
|
||||
<span class="blue">Max = 0.683ε (Mean = 0.0271ε)</span>
|
||||
</p>
|
||||
</td>
|
||||
<td>
|
||||
<p>
|
||||
<span class="blue">Max = 382ε (Mean = 22.2ε)</span>
|
||||
</p>
|
||||
</td>
|
||||
<td>
|
||||
<p>
|
||||
<span class="blue">Max = 315ε (Mean = 23.7ε)</span>
|
||||
</p>
|
||||
</td>
|
||||
</tr></tbody>
|
||||
</table></div>
|
||||
</div>
|
||||
<br class="table-break"><div class="table">
|
||||
<a name="math_toolkit.sf_beta.ibeta_inv_function.table_ibeta_invb"></a><p class="title"><b>Table 6.26. Error rates for ibeta_invb</b></p>
|
||||
<div class="table-contents"><table class="table" summary="Error rates for ibeta_invb">
|
||||
<colgroup>
|
||||
<col>
|
||||
<col>
|
||||
<col>
|
||||
<col>
|
||||
<col>
|
||||
</colgroup>
|
||||
<thead><tr>
|
||||
<th>
|
||||
</th>
|
||||
<th>
|
||||
<p>
|
||||
Microsoft Visual C++ version 12.0<br> Win32<br> double
|
||||
</p>
|
||||
</th>
|
||||
<th>
|
||||
<p>
|
||||
GNU C++ version 5.1.0<br> linux<br> double
|
||||
</p>
|
||||
</th>
|
||||
<th>
|
||||
<p>
|
||||
GNU C++ version 5.1.0<br> linux<br> long double
|
||||
</p>
|
||||
</th>
|
||||
<th>
|
||||
<p>
|
||||
Sun compiler version 0x5130<br> Sun Solaris<br> long double
|
||||
</p>
|
||||
</th>
|
||||
</tr></thead>
|
||||
<tbody><tr>
|
||||
<td>
|
||||
<p>
|
||||
Inverse incomplete beta
|
||||
</p>
|
||||
</td>
|
||||
<td>
|
||||
<p>
|
||||
<span class="blue">Max = 409ε (Mean = 17.9ε)</span>
|
||||
</p>
|
||||
</td>
|
||||
<td>
|
||||
<p>
|
||||
<span class="blue">Max = 0.836ε (Mean = 0.0491ε)</span>
|
||||
</p>
|
||||
</td>
|
||||
<td>
|
||||
<p>
|
||||
<span class="blue">Max = 407ε (Mean = 27.2ε)</span>
|
||||
</p>
|
||||
</td>
|
||||
<td>
|
||||
<p>
|
||||
<span class="blue">Max = 407ε (Mean = 24.4ε)</span>
|
||||
</p>
|
||||
</td>
|
||||
</tr></tbody>
|
||||
</table></div>
|
||||
</div>
|
||||
<br class="table-break"><div class="table">
|
||||
<a name="math_toolkit.sf_beta.ibeta_inv_function.table_ibetac_invb"></a><p class="title"><b>Table 6.27. Error rates for ibetac_invb</b></p>
|
||||
<div class="table-contents"><table class="table" summary="Error rates for ibetac_invb">
|
||||
<colgroup>
|
||||
<col>
|
||||
<col>
|
||||
<col>
|
||||
<col>
|
||||
<col>
|
||||
</colgroup>
|
||||
<thead><tr>
|
||||
<th>
|
||||
</th>
|
||||
<th>
|
||||
<p>
|
||||
Microsoft Visual C++ version 12.0<br> Win32<br> double
|
||||
</p>
|
||||
</th>
|
||||
<th>
|
||||
<p>
|
||||
GNU C++ version 5.1.0<br> linux<br> double
|
||||
</p>
|
||||
</th>
|
||||
<th>
|
||||
<p>
|
||||
GNU C++ version 5.1.0<br> linux<br> long double
|
||||
</p>
|
||||
</th>
|
||||
<th>
|
||||
<p>
|
||||
Sun compiler version 0x5130<br> Sun Solaris<br> long double
|
||||
</p>
|
||||
</th>
|
||||
</tr></thead>
|
||||
<tbody><tr>
|
||||
<td>
|
||||
<p>
|
||||
Inverse incomplete beta
|
||||
</p>
|
||||
</td>
|
||||
<td>
|
||||
<p>
|
||||
<span class="blue">Max = 329ε (Mean = 18.2ε)</span>
|
||||
</p>
|
||||
</td>
|
||||
<td>
|
||||
<p>
|
||||
<span class="blue">Max = 0.724ε (Mean = 0.0303ε)</span>
|
||||
</p>
|
||||
</td>
|
||||
<td>
|
||||
<p>
|
||||
<span class="blue">Max = 317ε (Mean = 19.7ε)</span>
|
||||
</p>
|
||||
</td>
|
||||
<td>
|
||||
<p>
|
||||
<span class="blue">Max = 369ε (Mean = 21.7ε)</span>
|
||||
</p>
|
||||
</td>
|
||||
</tr></tbody>
|
||||
</table></div>
|
||||
</div>
|
||||
<br class="table-break"><h5>
|
||||
<a name="math_toolkit.sf_beta.ibeta_inv_function.h2"></a>
|
||||
<span class="phrase"><a name="math_toolkit.sf_beta.ibeta_inv_function.testing"></a></span><a class="link" href="ibeta_inv_function.html#math_toolkit.sf_beta.ibeta_inv_function.testing">Testing</a>
|
||||
</h5>
|
||||
<p>
|
||||
There are two sets of tests:
|
||||
</p>
|
||||
<div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: disc; ">
|
||||
<li class="listitem">
|
||||
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 class="listitem">
|
||||
Accuracy checks using high precision test values. These measure the accuracy
|
||||
of the result, given exact input values.
|
||||
</li>
|
||||
</ul></div>
|
||||
<h5>
|
||||
<a name="math_toolkit.sf_beta.ibeta_inv_function.h3"></a>
|
||||
<span class="phrase"><a name="math_toolkit.sf_beta.ibeta_inv_function.implementation_of_ibeta_inv_and_"></a></span><a class="link" href="ibeta_inv_function.html#math_toolkit.sf_beta.ibeta_inv_function.implementation_of_ibeta_inv_and_">Implementation
|
||||
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 1/2 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 <span class="emphasis"><em>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.svg"></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.svg"></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.svg"></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.svg"></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.svg"></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.svg"></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.svg"></span>
|
||||
</p>
|
||||
<p>
|
||||
which can be improved somewhat to:
|
||||
</p>
|
||||
<p>
|
||||
<span class="inlinemediaobject"><img src="../../../equations/ibeta_inv1.svg"></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.svg"></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.svg"></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>
|
||||
<h5>
|
||||
<a name="math_toolkit.sf_beta.ibeta_inv_function.h4"></a>
|
||||
<span class="phrase"><a name="math_toolkit.sf_beta.ibeta_inv_function.implementation_of_inverses_on_th"></a></span><a class="link" href="ibeta_inv_function.html#math_toolkit.sf_beta.ibeta_inv_function.implementation_of_inverses_on_th">Implementation
|
||||
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 © 2006-2010, 2012-2014 Nikhar Agrawal,
|
||||
Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos, Hubert
|
||||
Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Johan Råde, Gautam Sewani,
|
||||
Benjamin Sobotta, Thijs van den Berg, Daryle Walker and Xiaogang Zhang<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">
|
||||
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|
||||
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|
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|
||||
</html>
|
||||
Reference in New Issue
Block a user