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<div class="titlepage"><div><div><h4 class="title">
<a name="math_toolkit.special.sf_gamma.gamma_ratios"></a><a class="link" href="gamma_ratios.html" title="Ratios of Gamma Functions"> Ratios
of Gamma Functions</a>
</h4></div></div></div>
<p>
</p>
<pre class="programlisting"><span class="preprocessor">#include</span> <span class="special"><</span><span class="identifier">boost</span><span class="special">/</span><span class="identifier">math</span><span class="special">/</span><span class="identifier">special_functions</span><span class="special">/</span><span class="identifier">gamma</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">></span>
</pre>
<p>
</p>
<pre class="programlisting"><span class="keyword">namespace</span> <span class="identifier">boost</span><span class="special">{</span> <span class="keyword">namespace</span> <span class="identifier">math</span><span class="special">{</span>
<span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">T1</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">T2</span><span class="special">></span>
<a class="link" href="../../main_overview/result_type.html" title="Calculation of the Type of the Result"><span class="emphasis"><em>calculated-result-type</em></span></a> <span class="identifier">tgamma_ratio</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="Policies">Policy</a><span class="special">></span>
<a class="link" href="../../main_overview/result_type.html" title="Calculation of the Type of the Result"><span class="emphasis"><em>calculated-result-type</em></span></a> <span class="identifier">tgamma_ratio</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="Policies">Policy</a><span class="special">&);</span>
<span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">T1</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">T2</span><span class="special">></span>
<a class="link" href="../../main_overview/result_type.html" title="Calculation of the Type of the Result"><span class="emphasis"><em>calculated-result-type</em></span></a> <span class="identifier">tgamma_delta_ratio</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">delta</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">Policy</span><span class="special">></span>
<a class="link" href="../../main_overview/result_type.html" title="Calculation of the Type of the Result"><span class="emphasis"><em>calculated-result-type</em></span></a> <span class="identifier">tgamma_delta_ratio</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">delta</span><span class="special">,</span> <span class="keyword">const</span> <a class="link" href="../../policy.html" title="Policies">Policy</a><span class="special">&);</span>
<span class="special">}}</span> <span class="comment">// namespaces
</span></pre>
<a name="math_toolkit.special.sf_gamma.gamma_ratios.description"></a><h5>
<a name="id1079859"></a>
<a class="link" href="gamma_ratios.html#math_toolkit.special.sf_gamma.gamma_ratios.description">Description</a>
</h5>
<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>
<a class="link" href="../../main_overview/result_type.html" title="Calculation of the Type of the Result"><span class="emphasis"><em>calculated-result-type</em></span></a> <span class="identifier">tgamma_ratio</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="Policies">Policy</a><span class="special">></span>
<a class="link" href="../../main_overview/result_type.html" title="Calculation of the Type of the Result"><span class="emphasis"><em>calculated-result-type</em></span></a> <span class="identifier">tgamma_ratio</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="Policies">Policy</a><span class="special">&);</span>
</pre>
<p>
Returns the ratio of gamma functions:
</p>
<p>
<span class="inlinemediaobject"><img src="../../../../equations/gamma_ratio0.png"></span>
</p>
<p>
</p>
<p>
The final <a class="link" href="../../policy.html" title="Policies">Policy</a> argument
is optional and can be used to control the behaviour of the function:
how it handles errors, what level of precision to use etc. Refer to the
<a class="link" href="../../policy.html" title="Policies">policy documentation for more details</a>.
</p>
<p>
</p>
<p>
Internally this just calls <code class="computeroutput"><span class="identifier">tgamma_delta_ratio</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">a</span><span class="special">)</span></code>.
</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>
<a class="link" href="../../main_overview/result_type.html" title="Calculation of the Type of the Result"><span class="emphasis"><em>calculated-result-type</em></span></a> <span class="identifier">tgamma_delta_ratio</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">delta</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="Policies">Policy</a><span class="special">></span>
<a class="link" href="../../main_overview/result_type.html" title="Calculation of the Type of the Result"><span class="emphasis"><em>calculated-result-type</em></span></a> <span class="identifier">tgamma_delta_ratio</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">delta</span><span class="special">,</span> <span class="keyword">const</span> <a class="link" href="../../policy.html" title="Policies">Policy</a><span class="special">&);</span>
</pre>
<p>
Returns the ratio of gamma functions:
</p>
<p>
<span class="inlinemediaobject"><img src="../../../../equations/gamma_ratio1.png"></span>
</p>
<p>
</p>
<p>
The final <a class="link" href="../../policy.html" title="Policies">Policy</a> argument
is optional and can be used to control the behaviour of the function:
how it handles errors, what level of precision to use etc. Refer to the
<a class="link" href="../../policy.html" title="Policies">policy documentation for more details</a>.
</p>
<p>
</p>
<p>
Note that the result is calculated accurately even when <span class="emphasis"><em>delta</em></span>
is small compared to <span class="emphasis"><em>a</em></span>: indeed even if <span class="emphasis"><em>a+delta
~ a</em></span>. The function is typically used when <span class="emphasis"><em>a</em></span>
is large and <span class="emphasis"><em>delta</em></span> is very small.
</p>
<p>
The return type of these functions is computed using the <a class="link" href="../../main_overview/result_type.html" title="Calculation of the Type of the Result"><span class="emphasis"><em>result
type calculation rules</em></span></a> when T1 and T2 are different types,
otherwise the result type is simple T1.
</p>
<p>
<span class="inlinemediaobject"><img src="../../../../graphs/tgamma_delta_ratio.png" align="middle"></span>
</p>
<a name="math_toolkit.special.sf_gamma.gamma_ratios.accuracy"></a><h5>
<a name="id1080416"></a>
<a class="link" href="gamma_ratios.html#math_toolkit.special.sf_gamma.gamma_ratios.accuracy">Accuracy</a>
</h5>
<p>
The following table shows the peak errors (in units of epsilon) found on
various platforms with various floating point types. Unless otherwise specified
any floating point type that is narrower than the one shown will have
<a class="link" href="../../backgrounders/relative_error.html#zero_error">effectively zero error</a>.
</p>
<div class="table">
<a name="id1080434"></a><p class="title"><b>Table 16. Errors In the Function tgamma_delta_ratio(a, delta)</b></p>
<div class="table-contents"><table class="table" summary="Errors In the Function tgamma_delta_ratio(a, delta)">
<colgroup>
<col>
<col>
<col>
</colgroup>
<thead><tr>
<th>
<p>
Significand Size
</p>
</th>
<th>
<p>
Platform and Compiler
</p>
</th>
<th>
<p>
20 < a < 80
</p>
<p>
and
</p>
<p>
delta < 1
</p>
</th>
</tr></thead>
<tbody>
<tr>
<td>
<p>
53
</p>
</td>
<td>
<p>
Win32, Visual C++ 8
</p>
</td>
<td>
<p>
Peak=16.9 Mean=1.7
</p>
</td>
</tr>
<tr>
<td>
<p>
64
</p>
</td>
<td>
<p>
Redhat Linux IA32, gcc-3.4.4
</p>
</td>
<td>
<p>
Peak=24 Mean=2.7
</p>
</td>
</tr>
<tr>
<td>
<p>
64
</p>
</td>
<td>
<p>
Redhat Linux IA64, gcc-3.4.4
</p>
</td>
<td>
<p>
Peak=12.8 Mean=1.8
</p>
</td>
</tr>
<tr>
<td>
<p>
113
</p>
</td>
<td>
<p>
HPUX IA64, aCC A.06.06
</p>
</td>
<td>
<p>
Peak=21.4 Mean=2.3
</p>
</td>
</tr>
</tbody>
</table></div>
</div>
<br class="table-break"><div class="table">
<a name="id1080598"></a><p class="title"><b>Table 17. Errors In the Function tgamma_ratio(a, b)</b></p>
<div class="table-contents"><table class="table" summary="Errors In the Function tgamma_ratio(a, b)">
<colgroup>
<col>
<col>
<col>
</colgroup>
<thead><tr>
<th>
<p>
Significand Size
</p>
</th>
<th>
<p>
Platform and Compiler
</p>
</th>
<th>
<p>
6 < a,b < 50
</p>
</th>
</tr></thead>
<tbody>
<tr>
<td>
<p>
53
</p>
</td>
<td>
<p>
Win32, Visual C++ 8
</p>
</td>
<td>
<p>
Peak=34 Mean=9
</p>
</td>
</tr>
<tr>
<td>
<p>
64
</p>
</td>
<td>
<p>
Redhat Linux IA32, gcc-3.4.4
</p>
</td>
<td>
<p>
Peak=91 Mean=23
</p>
</td>
</tr>
<tr>
<td>
<p>
64
</p>
</td>
<td>
<p>
Redhat Linux IA64, gcc-3.4.4
</p>
</td>
<td>
<p>
Peak=35.6 Mean=9.3
</p>
</td>
</tr>
<tr>
<td>
<p>
113
</p>
</td>
<td>
<p>
HPUX IA64, aCC A.06.06
</p>
</td>
<td>
<p>
Peak=43.9 Mean=13.2
</p>
</td>
</tr>
</tbody>
</table></div>
</div>
<br class="table-break"><a name="math_toolkit.special.sf_gamma.gamma_ratios.testing"></a><h5>
<a name="id1080755"></a>
<a class="link" href="gamma_ratios.html#math_toolkit.special.sf_gamma.gamma_ratios.testing">Testing</a>
</h5>
<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: about 300 decimal digits) and
a deliberately naive calculation of Γ(x)/Γ(y).
</p>
<a name="math_toolkit.special.sf_gamma.gamma_ratios.implementation"></a><h5>
<a name="id1080779"></a>
<a class="link" href="gamma_ratios.html#math_toolkit.special.sf_gamma.gamma_ratios.implementation">Implementation</a>
</h5>
<p>
The implementation of these functions is very similar to that of <a class="link" href="../sf_beta/beta_function.html" title="Beta">beta</a>, and is
based on combining similar power terms to improve accuracy and avoid spurious
overflow/underflow.
</p>
<p>
In addition there are optimisations for the situation where <span class="emphasis"><em>delta</em></span>
is a small integer: in which case this function is basically the reciprocal
of a rising factorial, or where both arguments are smallish integers: in
which case table lookup of factorials can be used to calculate the ratio.
</p>
</div>
<table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr>
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<td align="right"><div class="copyright-footer">Copyright © 2006 , 2007, 2008, 2009 John Maddock, Paul A. Bristow,
Hubert Holin, Xiaogang Zhang, Bruno Lalande, Johan Råde, Gautam Sewani
and Thijs van den Berg<p>
Distributed under the Boost Software License, Version 1.0. (See accompanying
file LICENSE_1_0.txt or copy at <a href="http://www.boost.org/LICENSE_1_0.txt" target="_top">http://www.boost.org/LICENSE_1_0.txt</a>)
</p>
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