<title>Log Gamma</title>
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<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 19. 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 19. 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> <span class="keyword">class</span> <a class="link" href="../../policy.html" title="Chapter 20. 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 20. 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 19. 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 19. 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> <span class="keyword">class</span> <a class="link" href="../../policy.html" title="Chapter 20. 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 20. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&);</span>
<span class="special">}}</span> <span class="comment">// namespaces</span>
</pre>
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>
+<div class="blockquote"><blockquote class="blockquote"><p>
+ <span class="inlinemediaobject"><img src="../../../equations/lgamm1.svg"></span>
+
+ </p></blockquote></div>
<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 19. Policies: Controlling Precision, Error Handling etc">Policy</a> argument is optional and can
+ The final <a class="link" href="../../policy.html" title="Chapter 20. 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 19. Policies: Controlling Precision, Error Handling etc">policy
+ what level of precision to use etc. Refer to the <a class="link" href="../../policy.html" title="Chapter 20. 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>
+<div class="blockquote"><blockquote class="blockquote"><p>
+ <span class="inlinemediaobject"><img src="../../../graphs/lgamma.svg" align="middle"></span>
+
+ </p></blockquote></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>: the result is of type <code class="computeroutput"><span class="keyword">double</span></code> if T is an integer type, or type T
error can be guaranteed.
</p>
<div class="table">
-<a name="math_toolkit.sf_gamma.lgamma.table_lgamma"></a><p class="title"><b>Table 7.3. Error rates for lgamma</b></p>
+<a name="math_toolkit.sf_gamma.lgamma.table_lgamma"></a><p class="title"><b>Table 8.3. Error rates for lgamma</b></p>
<div class="table-contents"><table class="table" summary="Error rates for lgamma">
<colgroup>
<col>
precision, and GCC-7.1/Ubuntu for <code class="computeroutput"><span class="keyword">long</span>
<span class="keyword">double</span></code> and <code class="computeroutput"><span class="identifier">__float128</span></code>.
</p>
-<p>
- <span class="inlinemediaobject"><img src="../../../graphs/lgamma__double.svg" align="middle"></span>
- </p>
-<p>
- <span class="inlinemediaobject"><img src="../../../graphs/lgamma__80_bit_long_double.svg" align="middle"></span>
- </p>
-<p>
- <span class="inlinemediaobject"><img src="../../../graphs/lgamma____float128.svg" align="middle"></span>
- </p>
+<div class="blockquote"><blockquote class="blockquote"><p>
+ <span class="inlinemediaobject"><img src="../../../graphs/lgamma__double.svg" align="middle"></span>
+
+ </p></blockquote></div>
+<div class="blockquote"><blockquote class="blockquote"><p>
+ <span class="inlinemediaobject"><img src="../../../graphs/lgamma__80_bit_long_double.svg" align="middle"></span>
+
+ </p></blockquote></div>
+<div class="blockquote"><blockquote class="blockquote"><p>
+ <span class="inlinemediaobject"><img src="../../../graphs/lgamma____float128.svg" align="middle"></span>
+
+ </p></blockquote></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>
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>
+<div class="blockquote"><blockquote class="blockquote"><p>
+ <span class="inlinemediaobject"><img src="../../../equations/gamma6.svg"></span>
+
+ </p></blockquote></div>
<p>
For small arguments, the logarithm of tgamma is used.
</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>
+<div class="blockquote"><blockquote class="blockquote"><p>
+ <span class="inlinemediaobject"><img src="../../../equations/lgamm3.svg"></span>
+
+ </p></blockquote></div>
<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>
+<div class="blockquote"><blockquote class="blockquote"><p>
+ <span class="inlinemediaobject"><img src="../../../equations/lgamm4.svg"></span>
+
+ </p></blockquote></div>
<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>.
+ 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>.
</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
+ so that its 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>
also be fed into log1p. Crucially, all of the terms tend to zero, as <span class="emphasis"><em>z
-> 1</em></span>:
</p>
+<div class="blockquote"><blockquote class="blockquote"><p>
+ <span class="inlinemediaobject"><img src="../../../equations/lgamm5.svg"></span>
+
+ </p></blockquote></div>
<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
+ 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>
+<div class="blockquote"><blockquote class="blockquote"><p>
+ <span class="inlinemediaobject"><img src="../../../equations/lgamm6.svg"></span>
+
+ </p></blockquote></div>
</div>
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