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<html>
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<title>Log Gamma</title>
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<div class="section">
<div class="titlepage"><div><div><h3 class="title">
<a name="math_toolkit.sf_gamma.lgamma"></a><a class="link" href="lgamma.html" title="Log Gamma">Log Gamma</a>
</h3></div></div></div>
<h5>
<a name="math_toolkit.sf_gamma.lgamma.h0"></a>
<span class="phrase"><a name="math_toolkit.sf_gamma.lgamma.synopsis"></a></span><a class="link" href="lgamma.html#math_toolkit.sf_gamma.lgamma.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">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="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">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="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">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="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">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="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.lgamma.h1"></a>
<span class="phrase"><a name="math_toolkit.sf_gamma.lgamma.description"></a></span><a class="link" href="lgamma.html#math_toolkit.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.svg"></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>
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>
<span class="inlinemediaobject"><img src="../../../graphs/lgamma.svg" 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="../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>: 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>
<h5>
<a name="math_toolkit.sf_gamma.lgamma.h2"></a>
<span class="phrase"><a name="math_toolkit.sf_gamma.lgamma.accuracy"></a></span><a class="link" href="lgamma.html#math_toolkit.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="../relative_error.html#math_toolkit.relative_error.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>
<h5>
<a name="math_toolkit.sf_gamma.lgamma.h3"></a>
<span class="phrase"><a name="math_toolkit.sf_gamma.lgamma.testing"></a></span><a class="link" href="lgamma.html#math_toolkit.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>
<h5>
<a name="math_toolkit.sf_gamma.lgamma.h4"></a>
<span class="phrase"><a name="math_toolkit.sf_gamma.lgamma.implementation"></a></span><a class="link" href="lgamma.html#math_toolkit.sf_gamma.lgamma.implementation">Implementation</a>
</h5>
<p>
The generic version of this function is implemented using Sterling's approximation
for large arguments:
</p>
<p>
<span class="inlinemediaobject"><img src="../../../equations/gamma6.svg"></span>
</p>
<p>
For small arguments, the logarithm of tgamma is used.
</p>
<p>
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.svg"></span>
</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. The logarithmic version of the
<a class="link" href="../lanczos.html" title="The Lanczos Approximation">Lanczos approximation</a> is:
</p>
<p>
<span class="inlinemediaobject"><img src="../../../equations/lgamm4.svg"></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="../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="../sf_implementation.html#math_toolkit.sf_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="../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="../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.svg"></span>
</p>
<p>
The C<sub>k</sub>   terms in the above are the same as in the <a class="link" href="../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.svg"></span>
</p>
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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>)
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