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<html>
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<div class="section">
<div class="titlepage"><div><div><h3 class="title">
<a name="math_toolkit.sf_gamma.tgamma"></a><a class="link" href="tgamma.html" title="Gamma">Gamma</a>
</h3></div></div></div>
<h5>
<a name="math_toolkit.sf_gamma.tgamma.h0"></a>
<span class="phrase"><a name="math_toolkit.sf_gamma.tgamma.synopsis"></a></span><a class="link" href="tgamma.html#math_toolkit.sf_gamma.tgamma.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">gamma</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">T</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">tgamma</span><span class="special">(</span><span class="identifier">T</span> <span class="identifier">z</span><span class="special">);</span>
<span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">,</span> <span class="keyword">class</span> <a class="link" href="../../policy.html" title="Chapter 14. 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">tgamma</span><span class="special">(</span><span class="identifier">T</span> <span class="identifier">z</span><span class="special">,</span> <span class="keyword">const</span> <a class="link" href="../../policy.html" title="Chapter 14. 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">T</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">tgamma1pm1</span><span class="special">(</span><span class="identifier">T</span> <span class="identifier">dz</span><span class="special">);</span>
<span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">,</span> <span class="keyword">class</span> <a class="link" href="../../policy.html" title="Chapter 14. 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">tgamma1pm1</span><span class="special">(</span><span class="identifier">T</span> <span class="identifier">dz</span><span class="special">,</span> <span class="keyword">const</span> <a class="link" href="../../policy.html" title="Chapter 14. 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_gamma.tgamma.h1"></a>
<span class="phrase"><a name="math_toolkit.sf_gamma.tgamma.description"></a></span><a class="link" href="tgamma.html#math_toolkit.sf_gamma.tgamma.description">Description</a>
</h5>
<pre class="programlisting"><span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">T</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">tgamma</span><span class="special">(</span><span class="identifier">T</span> <span class="identifier">z</span><span class="special">);</span>
<span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">,</span> <span class="keyword">class</span> <a class="link" href="../../policy.html" title="Chapter 14. 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">tgamma</span><span class="special">(</span><span class="identifier">T</span> <span class="identifier">z</span><span class="special">,</span> <span class="keyword">const</span> <a class="link" href="../../policy.html" title="Chapter 14. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&);</span>
</pre>
<p>
Returns the "true gamma" (hence name tgamma) of value z:
</p>
<p>
<span class="inlinemediaobject"><img src="../../../equations/gamm1.png"></span>
</p>
<p>
<span class="inlinemediaobject"><img src="../../../graphs/tgamma.png" align="middle"></span>
</p>
<p>
The final <a class="link" href="../../policy.html" title="Chapter 14. 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 14. Policies: Controlling Precision, Error Handling etc">policy
documentation for more details</a>.
</p>
<p>
There are effectively two versions of the <a href="http://en.wikipedia.org/wiki/Gamma_function" target="_top">tgamma</a>
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 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>: the result is <code class="computeroutput"><span class="keyword">double</span></code>
when T is an integer type, and T otherwise.
</p>
<pre class="programlisting"><span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">T</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">tgamma1pm1</span><span class="special">(</span><span class="identifier">T</span> <span class="identifier">dz</span><span class="special">);</span>
<span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">,</span> <span class="keyword">class</span> <a class="link" href="../../policy.html" title="Chapter 14. 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">tgamma1pm1</span><span class="special">(</span><span class="identifier">T</span> <span class="identifier">dz</span><span class="special">,</span> <span class="keyword">const</span> <a class="link" href="../../policy.html" title="Chapter 14. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&);</span>
</pre>
<p>
Returns <code class="computeroutput"><span class="identifier">tgamma</span><span class="special">(</span><span class="identifier">dz</span> <span class="special">+</span> <span class="number">1</span><span class="special">)</span> <span class="special">-</span> <span class="number">1</span></code>.
Internally the implementation does not make use of the addition and subtraction
implied by the definition, leading to accurate results even for very small
<code class="computeroutput"><span class="identifier">dz</span></code>. However, the implementation
is capped to either 35 digit accuracy, or to the precision of the <a class="link" href="../lanczos.html" title="The Lanczos Approximation">Lanczos
approximation</a> associated with type T, whichever is more accurate.
</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>: the result is <code class="computeroutput"><span class="keyword">double</span></code>
when T is an integer type, and T otherwise.
</p>
<p>
The final <a class="link" href="../../policy.html" title="Chapter 14. 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 14. Policies: Controlling Precision, Error Handling etc">policy
documentation for more details</a>.
</p>
<h5>
<a name="math_toolkit.sf_gamma.tgamma.h2"></a>
<span class="phrase"><a name="math_toolkit.sf_gamma.tgamma.accuracy"></a></span><a class="link" href="tgamma.html#math_toolkit.sf_gamma.tgamma.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, along with comparisons
to the <a href="http://www.gnu.org/software/gsl/" target="_top">GSL-1.9</a>, <a href="http://www.gnu.org/software/libc/" target="_top">GNU C Lib</a>, <a href="http://docs.hp.com/en/B9106-90010/index.html" target="_top">HP-UX
C Library</a> and <a href="http://www.netlib.org/cephes/" target="_top">Cephes</a>
libraries. Unless otherwise specified any floating point type that is narrower
than the one shown will have <a class="link" href="../relative_error.html#math_toolkit.relative_error.zero_error">effectively
zero error</a>.
</p>
<div class="informaltable"><table class="table">
<colgroup>
<col>
<col>
<col>
<col>
<col>
<col>
</colgroup>
<thead><tr>
<th>
<p>
Significand Size
</p>
</th>
<th>
<p>
Platform and Compiler
</p>
</th>
<th>
<p>
Factorials and Half factorials
</p>
</th>
<th>
<p>
Values Near Zero
</p>
</th>
<th>
<p>
Values Near 1 or 2
</p>
</th>
<th>
<p>
Values Near a Negative Pole
</p>
</th>
</tr></thead>
<tbody>
<tr>
<td>
<p>
53
</p>
</td>
<td>
<p>
Win32 Visual C++ 8
</p>
</td>
<td>
<p>
Peak=1.9 Mean=0.7
</p>
<p>
(GSL=3.9)
</p>
<p>
(<a href="http://www.netlib.org/cephes/" target="_top">Cephes</a>=3.0)
</p>
</td>
<td>
<p>
Peak=2.0 Mean=1.1
</p>
<p>
(GSL=4.5)
</p>
<p>
(<a href="http://www.netlib.org/cephes/" target="_top">Cephes</a>=1)
</p>
</td>
<td>
<p>
Peak=2.0 Mean=1.1
</p>
<p>
(GSL=7.9)
</p>
<p>
(<a href="http://www.netlib.org/cephes/" target="_top">Cephes</a>=1.0)
</p>
</td>
<td>
<p>
Peak=2.6 Mean=1.3
</p>
<p>
(GSL=2.5)
</p>
<p>
(<a href="http://www.netlib.org/cephes/" target="_top">Cephes</a>=2.7)
</p>
</td>
</tr>
<tr>
<td>
<p>
64
</p>
</td>
<td>
<p>
Linux IA32 / GCC
</p>
</td>
<td>
<p>
Peak=300 Mean=49.5
</p>
<p>
(<a href="http://www.gnu.org/software/libc/" target="_top">GNU C Lib</a>
Peak=395 Mean=89)
</p>
</td>
<td>
<p>
Peak=3.0 Mean=1.4
</p>
<p>
(<a href="http://www.gnu.org/software/libc/" target="_top">GNU C Lib</a>
Peak=11 Mean=3.3)
</p>
</td>
<td>
<p>
Peak=5.0 Mean=1.8
</p>
<p>
(<a href="http://www.gnu.org/software/libc/" target="_top">GNU C Lib</a>
Peak=0.92 Mean=0.2)
</p>
</td>
<td>
<p>
Peak=157 Mean=65
</p>
<p>
(<a href="http://www.gnu.org/software/libc/" target="_top">GNU C Lib</a>
Peak=205 Mean=108)
</p>
</td>
</tr>
<tr>
<td>
<p>
64
</p>
</td>
<td>
<p>
Linux IA64 / GCC
</p>
</td>
<td>
<p>
<a href="http://www.gnu.org/software/libc/" target="_top">GNU C Lib</a>
Peak 2.8 Mean=0.9
</p>
<p>
(<a href="http://www.gnu.org/software/libc/" target="_top">GNU C Lib</a>
Peak 0.7)
</p>
</td>
<td>
<p>
Peak=4.8 Mean=1.5
</p>
<p>
(<a href="http://www.gnu.org/software/libc/" target="_top">GNU C Lib</a>
Peak 0)
</p>
</td>
<td>
<p>
Peak=4.8 Mean=1.5
</p>
<p>
(<a href="http://www.gnu.org/software/libc/" target="_top">GNU C Lib</a>
Peak 0)
</p>
</td>
<td>
<p>
Peak=5.0 Mean=1.7 (<a href="http://www.gnu.org/software/libc/" target="_top">GNU
C Lib</a> Peak 0)
</p>
</td>
</tr>
<tr>
<td>
<p>
113
</p>
</td>
<td>
<p>
HPUX IA64, aCC A.06.06
</p>
</td>
<td>
<p>
Peak=2.5 Mean=1.1
</p>
<p>
(<a href="http://docs.hp.com/en/B9106-90010/index.html" target="_top">HP-UX
C Library</a> Peak 0)
</p>
</td>
<td>
<p>
Peak=3.5 Mean=1.7
</p>
<p>
(<a href="http://docs.hp.com/en/B9106-90010/index.html" target="_top">HP-UX
C Library</a> Peak 0)
</p>
</td>
<td>
<p>
Peak=3.5 Mean=1.6
</p>
<p>
(<a href="http://docs.hp.com/en/B9106-90010/index.html" target="_top">HP-UX
C Library</a> Peak 0)
</p>
</td>
<td>
<p>
Peak=5.2 Mean=1.92
</p>
<p>
(<a href="http://docs.hp.com/en/B9106-90010/index.html" target="_top">HP-UX
C Library</a> Peak 0)
</p>
</td>
</tr>
</tbody>
</table></div>
<h5>
<a name="math_toolkit.sf_gamma.tgamma.h3"></a>
<span class="phrase"><a name="math_toolkit.sf_gamma.tgamma.testing"></a></span><a class="link" href="tgamma.html#math_toolkit.sf_gamma.tgamma.testing">Testing</a>
</h5>
<p>
The gamma is relatively easy to test: factorials and half-integer factorials
can be calculated exactly by other means and compared with the gamma function.
In addition, some accuracy tests in known tricky areas were computed at high
precision using the generic version of this function.
</p>
<p>
The function <code class="computeroutput"><span class="identifier">tgamma1pm1</span></code> is
tested against values calculated very naively using the formula <code class="computeroutput"><span class="identifier">tgamma</span><span class="special">(</span><span class="number">1</span><span class="special">+</span><span class="identifier">dz</span><span class="special">)-</span><span class="number">1</span></code> with a
lanczos approximation accurate to around 100 decimal digits.
</p>
<h5>
<a name="math_toolkit.sf_gamma.tgamma.h4"></a>
<span class="phrase"><a name="math_toolkit.sf_gamma.tgamma.implementation"></a></span><a class="link" href="tgamma.html#math_toolkit.sf_gamma.tgamma.implementation">Implementation</a>
</h5>
<p>
The generic version of the <code class="computeroutput"><span class="identifier">tgamma</span></code>
function is implemented Sterling's approximation for lgamma for large z:
</p>
<p>
<span class="inlinemediaobject"><img src="../../../equations/gamma6.png"></span>
</p>
<p>
Following exponentiation, downward recursion is then used for small values
of z.
</p>
<p>
For types of known precision the <a class="link" href="../lanczos.html" title="The Lanczos Approximation">Lanczos
approximation</a> is used, a traits class <code class="computeroutput"><span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">::</span><span class="identifier">lanczos</span><span class="special">::</span><span class="identifier">lanczos_traits</span></code>
maps type T to an appropriate approximation.
</p>
<p>
For z in the range -20 < z < 1 then recursion is used to shift to z
> 1 via:
</p>
<p>
<span class="inlinemediaobject"><img src="../../../equations/gamm3.png"></span>
</p>
<p>
For very small z, this helps to preserve the identity:
</p>
<p>
<span class="inlinemediaobject"><img src="../../../equations/gamm4.png"></span>
</p>
<p>
For z < -20 the reflection formula:
</p>
<p>
<span class="inlinemediaobject"><img src="../../../equations/gamm5.png"></span>
</p>
<p>
is used. Particular care has to be taken to evaluate the <code class="literal">z * sin(π   *
z)</code> part: a special routine is used to reduce z prior to multiplying
by π   to ensure that the result in is the range [0, π/2]. Without this an excessive
amount of error occurs in this region (which is hard enough already, as the
rate of change near a negative pole is <span class="emphasis"><em>exceptionally</em></span>
high).
</p>
<p>
Finally if the argument is a small integer then table lookup of the factorial
is used.
</p>
<p>
The function <code class="computeroutput"><span class="identifier">tgamma1pm1</span></code> is
implemented using rational approximations <a class="link" href="../sf_implementation.html#math_toolkit.sf_implementation.rational_approximations_used">devised
by JM</a> in the region <code class="computeroutput"><span class="special">-</span><span class="number">0.5</span> <span class="special"><</span> <span class="identifier">dz</span>
<span class="special"><</span> <span class="number">2</span></code>.
These are the same approximations (and internal routines) that are used for
<a class="link" href="lgamma.html" title="Log Gamma">lgamma</a>, and so aren't
detailed further here. The result of the approximation is <code class="computeroutput"><span class="identifier">log</span><span class="special">(</span><span class="identifier">tgamma</span><span class="special">(</span><span class="identifier">dz</span><span class="special">+</span><span class="number">1</span><span class="special">))</span></code> which can
fed into <a class="link" href="../powers/expm1.html" title="expm1">expm1</a> to give the
desired result. Outside the range <code class="computeroutput"><span class="special">-</span><span class="number">0.5</span> <span class="special"><</span> <span class="identifier">dz</span>
<span class="special"><</span> <span class="number">2</span></code>
then the naive formula <code class="computeroutput"><span class="identifier">tgamma1pm1</span><span class="special">(</span><span class="identifier">dz</span><span class="special">)</span>
<span class="special">=</span> <span class="identifier">tgamma</span><span class="special">(</span><span class="identifier">dz</span><span class="special">+</span><span class="number">1</span><span class="special">)-</span><span class="number">1</span></code>
can be used directly.
</p>
</div>
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<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, 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>
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