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<div class="titlepage"><div><div><h4 class="title">
<a name="math_toolkit.special.sf_gamma.lgamma"></a><a class="link" href="lgamma.html" title="Log Gamma"> Log Gamma</a>
</h4></div></div></div>
<a name="math_toolkit.special.sf_gamma.lgamma.synopsis"></a><h5>
<a name="id1073809"></a>
<a class="link" href="lgamma.html#math_toolkit.special.sf_gamma.lgamma.synopsis">Synopsis</a>
</h5>
<p>
</p>
<pre class="programlisting"><span class="preprocessor">#include</span> <span class="special"><</span><span class="identifier">boost</span><span class="special">/</span><span class="identifier">math</span><span class="special">/</span><span class="identifier">special_functions</span><span class="special">/</span><span class="identifier">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">T</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">lgamma</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="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">lgamma</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="Policies">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="../../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">lgamma</span><span class="special">(</span><span class="identifier">T</span> <span class="identifier">z</span><span class="special">,</span> <span class="keyword">int</span><span class="special">*</span> <span class="identifier">sign</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="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">lgamma</span><span class="special">(</span><span class="identifier">T</span> <span class="identifier">z</span><span class="special">,</span> <span class="keyword">int</span><span class="special">*</span> <span class="identifier">sign</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.lgamma.description"></a><h5>
<a name="id1074192"></a>
<a class="link" href="lgamma.html#math_toolkit.special.sf_gamma.lgamma.description">Description</a>
</h5>
<p>
The <a href="http://en.wikipedia.org/wiki/Gamma_function" target="_top">lgamma function</a>
is defined by:
</p>
<p>
<span class="inlinemediaobject"><img src="../../../../equations/lgamm1.png"></span>
</p>
<p>
The second form of the function takes a pointer to an integer, which if
non-null is set on output to the sign of tgamma(z).
</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>
<span class="inlinemediaobject"><img src="../../../../graphs/lgamma.png" align="middle"></span>
</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="../../backgrounders/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="../../backgrounders/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="../../backgrounders/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="../../main_overview/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 of type <code class="computeroutput"><span class="keyword">double</span></code> if T is an integer type, or type
T otherwise.
</p>
<a name="math_toolkit.special.sf_gamma.lgamma.accuracy"></a><h5>
<a name="id1075445"></a>
<a class="link" href="lgamma.html#math_toolkit.special.sf_gamma.lgamma.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="../../backgrounders/relative_error.html#zero_error">effectively zero error</a>.
</p>
<p>
Note that while the relative errors near the positive roots of lgamma are
very low, the lgamma function has an infinite number of irrational roots
for negative arguments: very close to these negative roots only a low absolute
error can be guaranteed.
</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=0.88 Mean=0.14
</p>
<p>
(GSL=33) (<a href="http://www.netlib.org/cephes/" target="_top">Cephes</a>=1.5)
</p>
</td>
<td>
<p>
Peak=0.96 Mean=0.46
</p>
<p>
(GSL=5.2) (<a href="http://www.netlib.org/cephes/" target="_top">Cephes</a>=1.1)
</p>
</td>
<td>
<p>
Peak=0.86 Mean=0.46
</p>
<p>
(GSL=1168) (<a href="http://www.netlib.org/cephes/" target="_top">Cephes</a>~500000)
</p>
</td>
<td>
<p>
Peak=4.2 Mean=1.3
</p>
<p>
(GSL=25) (<a href="http://www.netlib.org/cephes/" target="_top">Cephes</a>=1.6)
</p>
</td>
</tr>
<tr>
<td>
<p>
64
</p>
</td>
<td>
<p>
Linux IA32 / GCC
</p>
</td>
<td>
<p>
Peak=1.9 Mean=0.43
</p>
<p>
(<a href="http://www.gnu.org/software/libc/" target="_top">GNU C Lib</a>
Peak=1.7 Mean=0.49)
</p>
</td>
<td>
<p>
Peak=1.4 Mean=0.57
</p>
<p>
(<a href="http://www.gnu.org/software/libc/" target="_top">GNU C Lib</a>
Peak= 0.96 Mean=0.54)
</p>
</td>
<td>
<p>
Peak=0.86 Mean=0.35
</p>
<p>
(<a href="http://www.gnu.org/software/libc/" target="_top">GNU C Lib</a>
Peak=0.74 Mean=0.26)
</p>
</td>
<td>
<p>
Peak=6.0 Mean=1.8
</p>
<p>
(<a href="http://www.gnu.org/software/libc/" target="_top">GNU C Lib</a>
Peak=3.0 Mean=0.86)
</p>
</td>
</tr>
<tr>
<td>
<p>
64
</p>
</td>
<td>
<p>
Linux IA64 / GCC
</p>
</td>
<td>
<p>
Peak=0.99 Mean=0.12
</p>
<p>
(<a href="http://www.gnu.org/software/libc/" target="_top">GNU C Lib</a>
Peak 0)
</p>
</td>
<td>
<p>
Pek=1.2 Mean=0.6
</p>
<p>
(<a href="http://www.gnu.org/software/libc/" target="_top">GNU C Lib</a>
Peak 0)
</p>
</td>
<td>
<p>
Peak=0.86 Mean=0.16
</p>
<p>
(<a href="http://www.gnu.org/software/libc/" target="_top">GNU C Lib</a>
Peak 0)
</p>
</td>
<td>
<p>
Peak=2.3 Mean=0.69
</p>
<p>
(<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=0.96 Mean=0.13
</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=0.99 Mean=0.53
</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=0.9 Mean=0.4
</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=3.0 Mean=0.9
</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>
<a name="math_toolkit.special.sf_gamma.lgamma.testing"></a><h5>
<a name="id1075908"></a>
<a class="link" href="lgamma.html#math_toolkit.special.sf_gamma.lgamma.testing">Testing</a>
</h5>
<p>
The main tests for this function involve comparisons against the logs of
the factorials which can be independently calculated to very high accuracy.
</p>
<p>
Random tests in key problem areas are also used.
</p>
<a name="math_toolkit.special.sf_gamma.lgamma.implementation"></a><h5>
<a name="id1075928"></a>
<a class="link" href="lgamma.html#math_toolkit.special.sf_gamma.lgamma.implementation">Implementation</a>
</h5>
<p>
The generic version of this function is implemented by combining the series
and continued fraction representations for the incomplete gamma function:
</p>
<p>
<span class="inlinemediaobject"><img src="../../../../equations/lgamm2.png"></span>
</p>
<p>
where <span class="emphasis"><em>l</em></span> is an arbitrary integration limit: choosing
<code class="literal">l = max(10, a)</code> seems to work fairly well. For negative
<span class="emphasis"><em>z</em></span> the logarithm version of the reflection formula
is used:
</p>
<p>
<span class="inlinemediaobject"><img src="../../../../equations/lgamm3.png"></span>
</p>
<p>
For types of known precision, the <a class="link" href="../../backgrounders/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. The logarithmic version of
the <a class="link" href="../../backgrounders/lanczos.html" title="The Lanczos Approximation">Lanczos approximation</a>
is:
</p>
<p>
<span class="inlinemediaobject"><img src="../../../../equations/lgamm4.png"></span>
</p>
<p>
Where L<sub>e,g</sub> is the Lanczos sum, scaled by e<sup>g</sup>.
</p>
<p>
As before the reflection formula is used for <span class="emphasis"><em>z < 0</em></span>.
</p>
<p>
When z is very near 1 or 2, then the logarithmic version of the <a class="link" href="../../backgrounders/lanczos.html" title="The Lanczos Approximation">Lanczos
approximation</a> suffers very badly from cancellation error: indeed
for values sufficiently close to 1 or 2, arbitrarily large relative errors
can be obtained (even though the absolute error is tiny).
</p>
<p>
For types with up to 113 bits of precision (up to and including 128-bit
long doubles), root-preserving rational approximations <a class="link" href="../../backgrounders/implementation.html#math_toolkit.backgrounders.implementation.rational_approximations_used">devised
by JM</a> are used over the intervals [1,2] and [2,3]. Over the interval
[2,3] the approximation form used is:
</p>
<pre class="programlisting"><span class="identifier">lgamma</span><span class="special">(</span><span class="identifier">z</span><span class="special">)</span> <span class="special">=</span> <span class="special">(</span><span class="identifier">z</span><span class="special">-</span><span class="number">2</span><span class="special">)(</span><span class="identifier">z</span><span class="special">+</span><span class="number">1</span><span class="special">)(</span><span class="identifier">Y</span> <span class="special">+</span> <span class="identifier">R</span><span class="special">(</span><span class="identifier">z</span><span class="special">-</span><span class="number">2</span><span class="special">));</span>
</pre>
<p>
Where Y is a constant, and R(z-2) is the rational approximation: optimised
so that it's absolute error is tiny compared to Y. In addition small values
of z greater than 3 can handled by argument reduction using the recurrence
relation:
</p>
<pre class="programlisting"><span class="identifier">lgamma</span><span class="special">(</span><span class="identifier">z</span><span class="special">+</span><span class="number">1</span><span class="special">)</span> <span class="special">=</span> <span class="identifier">log</span><span class="special">(</span><span class="identifier">z</span><span class="special">)</span> <span class="special">+</span> <span class="identifier">lgamma</span><span class="special">(</span><span class="identifier">z</span><span class="special">);</span>
</pre>
<p>
Over the interval [1,2] two approximations have to be used, one for small
z uses:
</p>
<pre class="programlisting"><span class="identifier">lgamma</span><span class="special">(</span><span class="identifier">z</span><span class="special">)</span> <span class="special">=</span> <span class="special">(</span><span class="identifier">z</span><span class="special">-</span><span class="number">1</span><span class="special">)(</span><span class="identifier">z</span><span class="special">-</span><span class="number">2</span><span class="special">)(</span><span class="identifier">Y</span> <span class="special">+</span> <span class="identifier">R</span><span class="special">(</span><span class="identifier">z</span><span class="special">-</span><span class="number">1</span><span class="special">));</span>
</pre>
<p>
Once again Y is a constant, and R(z-1) is optimised for low absolute error
compared to Y. For z > 1.5 the above form wouldn't converge to a minimax
solution but this similar form does:
</p>
<pre class="programlisting"><span class="identifier">lgamma</span><span class="special">(</span><span class="identifier">z</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">z</span><span class="special">)(</span><span class="number">1</span><span class="special">-</span><span class="identifier">z</span><span class="special">)(</span><span class="identifier">Y</span> <span class="special">+</span> <span class="identifier">R</span><span class="special">(</span><span class="number">2</span><span class="special">-</span><span class="identifier">z</span><span class="special">));</span>
</pre>
<p>
Finally for z < 1 the recurrence relation can be used to move to z >
1:
</p>
<pre class="programlisting"><span class="identifier">lgamma</span><span class="special">(</span><span class="identifier">z</span><span class="special">)</span> <span class="special">=</span> <span class="identifier">lgamma</span><span class="special">(</span><span class="identifier">z</span><span class="special">+</span><span class="number">1</span><span class="special">)</span> <span class="special">-</span> <span class="identifier">log</span><span class="special">(</span><span class="identifier">z</span><span class="special">);</span>
</pre>
<p>
Note that while this involves a subtraction, it appears not to suffer from
cancellation error: as z decreases from 1 the <code class="computeroutput"><span class="special">-</span><span class="identifier">log</span><span class="special">(</span><span class="identifier">z</span><span class="special">)</span></code> term
grows positive much more rapidly than the <code class="computeroutput"><span class="identifier">lgamma</span><span class="special">(</span><span class="identifier">z</span><span class="special">+</span><span class="number">1</span><span class="special">)</span></code> term becomes
negative. So in this specific case, significant digits are preserved, rather
than cancelled.
</p>
<p>
For other types which do have a <a class="link" href="../../backgrounders/lanczos.html" title="The Lanczos Approximation">Lanczos
approximation</a> defined for them the current solution is as follows:
imagine we balance the two terms in the <a class="link" href="../../backgrounders/lanczos.html" title="The Lanczos Approximation">Lanczos
approximation</a> by dividing the power term by its value at <span class="emphasis"><em>z
= 1</em></span>, and then multiplying the Lanczos coefficients by the same
value. Now each term will take the value 1 at <span class="emphasis"><em>z = 1</em></span>
and we can rearrange the power terms in terms of log1p. Likewise if we
subtract 1 from the Lanczos sum part (algebraically, by subtracting the
value of each term at <span class="emphasis"><em>z = 1</em></span>), we obtain a new summation
that can be also be fed into log1p. Crucially, all of the terms tend to
zero, as <span class="emphasis"><em>z -> 1</em></span>:
</p>
<p>
<span class="inlinemediaobject"><img src="../../../../equations/lgamm5.png"></span>
</p>
<p>
The C<sub>k</sub> terms in the above are the same as in the <a class="link" href="../../backgrounders/lanczos.html" title="The Lanczos Approximation">Lanczos
approximation</a>.
</p>
<p>
A similar rearrangement can be performed at <span class="emphasis"><em>z = 2</em></span>:
</p>
<p>
<span class="inlinemediaobject"><img src="../../../../equations/lgamm6.png"></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 , 2007, 2008, 2009 John Maddock, Paul A. Bristow,
Hubert Holin, Xiaogang Zhang, Bruno Lalande, Johan Råde, Gautam Sewani
and Thijs van den Berg<p>
Distributed under the Boost Software License, Version 1.0. (See accompanying
file LICENSE_1_0.txt or copy at <a href="http://www.boost.org/LICENSE_1_0.txt" target="_top">http://www.boost.org/LICENSE_1_0.txt</a>)
</p>
</div></td>
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