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<div class="section">
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<div class="titlepage"><div><div><h3 class="title">
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<a name="math_toolkit.sf_gamma.lgamma"></a><a class="link" href="lgamma.html" title="Log Gamma">Log Gamma</a>
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</h3></div></div></div>
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<h5>
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<a name="math_toolkit.sf_gamma.lgamma.h0"></a>
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<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>
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</h5>
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<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>
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</pre>
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<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>
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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>
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<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>
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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> <span class="keyword">class</span> <a class="link" href="../../policy.html" title="Chapter 13. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">></span>
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<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 13. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&);</span>
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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>
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<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>
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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> <span class="keyword">class</span> <a class="link" href="../../policy.html" title="Chapter 13. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">></span>
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<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 13. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&);</span>
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<span class="special">}}</span> <span class="comment">// namespaces</span>
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</pre>
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<h5>
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<a name="math_toolkit.sf_gamma.lgamma.h1"></a>
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<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>
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</h5>
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<p>
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The <a href="http://en.wikipedia.org/wiki/Gamma_function" target="_top">lgamma function</a>
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is defined by:
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</p>
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<p>
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<span class="inlinemediaobject"><img src="../../../equations/lgamm1.png"></span>
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</p>
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<p>
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The second form of the function takes a pointer to an integer, which if non-null
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is set on output to the sign of tgamma(z).
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</p>
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<p>
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The final <a class="link" href="../../policy.html" title="Chapter 13. Policies: Controlling Precision, Error Handling etc">Policy</a> argument is optional and can
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be used to control the behaviour of the function: how it handles errors,
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what level of precision to use etc. Refer to the <a class="link" href="../../policy.html" title="Chapter 13. Policies: Controlling Precision, Error Handling etc">policy
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documentation for more details</a>.
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</p>
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<p>
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<span class="inlinemediaobject"><img src="../../../graphs/lgamma.png" align="middle"></span>
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</p>
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<p>
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There are effectively two versions of this function internally: a fully generic
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version that is slow, but reasonably accurate, and a much more efficient
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approximation that is used where the number of digits in the significand
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of T correspond to a certain <a class="link" href="../lanczos.html" title="The Lanczos Approximation">Lanczos
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approximation</a>. In practice, any built-in floating-point type you will
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encounter has an appropriate <a class="link" href="../lanczos.html" title="The Lanczos Approximation">Lanczos
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approximation</a> defined for it. It is also possible, given enough machine
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time, to generate further <a class="link" href="../lanczos.html" title="The Lanczos Approximation">Lanczos approximation</a>'s
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using the program libs/math/tools/lanczos_generator.cpp.
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</p>
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<p>
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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
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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
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otherwise.
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</p>
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<h5>
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<a name="math_toolkit.sf_gamma.lgamma.h2"></a>
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<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>
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</h5>
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<p>
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The following table shows the peak errors (in units of epsilon) found on
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various platforms with various floating point types, along with comparisons
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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
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C Library</a> and <a href="http://www.netlib.org/cephes/" target="_top">Cephes</a>
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libraries. Unless otherwise specified any floating point type that is narrower
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than the one shown will have <a class="link" href="../relative_error.html#math_toolkit.relative_error.zero_error">effectively
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zero error</a>.
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</p>
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<p>
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Note that while the relative errors near the positive roots of lgamma are
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very low, the lgamma function has an infinite number of irrational roots
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for negative arguments: very close to these negative roots only a low absolute
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error can be guaranteed.
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</p>
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<div class="informaltable"><table class="table">
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<colgroup>
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<col>
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<col>
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<col>
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<col>
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<col>
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<col>
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</colgroup>
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<thead><tr>
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<th>
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<p>
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Significand Size
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</p>
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</th>
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<th>
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<p>
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Platform and Compiler
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</p>
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</th>
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<th>
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<p>
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Factorials and Half factorials
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</p>
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</th>
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<th>
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<p>
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Values Near Zero
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</p>
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</th>
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<th>
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<p>
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Values Near 1 or 2
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</p>
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</th>
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<th>
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<p>
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Values Near a Negative Pole
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</p>
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</th>
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</tr></thead>
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<tbody>
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<tr>
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<td>
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<p>
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53
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</p>
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</td>
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<td>
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<p>
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Win32 Visual C++ 8
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</p>
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</td>
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<td>
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<p>
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Peak=0.88 Mean=0.14
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</p>
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<p>
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(GSL=33) (<a href="http://www.netlib.org/cephes/" target="_top">Cephes</a>=1.5)
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</p>
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</td>
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<td>
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<p>
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Peak=0.96 Mean=0.46
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</p>
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<p>
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(GSL=5.2) (<a href="http://www.netlib.org/cephes/" target="_top">Cephes</a>=1.1)
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</p>
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</td>
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<td>
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<p>
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Peak=0.86 Mean=0.46
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</p>
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<p>
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(GSL=1168) (<a href="http://www.netlib.org/cephes/" target="_top">Cephes</a>~500000)
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</p>
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</td>
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<td>
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<p>
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Peak=4.2 Mean=1.3
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</p>
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<p>
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(GSL=25) (<a href="http://www.netlib.org/cephes/" target="_top">Cephes</a>=1.6)
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</p>
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</td>
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</tr>
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<tr>
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<td>
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<p>
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64
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</p>
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</td>
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<td>
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<p>
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Linux IA32 / GCC
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</p>
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</td>
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<td>
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<p>
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Peak=1.9 Mean=0.43
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</p>
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<p>
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(<a href="http://www.gnu.org/software/libc/" target="_top">GNU C Lib</a>
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Peak=1.7 Mean=0.49)
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</p>
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</td>
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<td>
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<p>
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Peak=1.4 Mean=0.57
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</p>
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<p>
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(<a href="http://www.gnu.org/software/libc/" target="_top">GNU C Lib</a>
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Peak= 0.96 Mean=0.54)
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</p>
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</td>
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<td>
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<p>
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Peak=0.86 Mean=0.35
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</p>
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<p>
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(<a href="http://www.gnu.org/software/libc/" target="_top">GNU C Lib</a>
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Peak=0.74 Mean=0.26)
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</p>
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</td>
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<td>
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<p>
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Peak=6.0 Mean=1.8
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</p>
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<p>
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(<a href="http://www.gnu.org/software/libc/" target="_top">GNU C Lib</a>
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Peak=3.0 Mean=0.86)
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</p>
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</td>
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</tr>
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<tr>
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<td>
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<p>
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64
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</p>
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</td>
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<td>
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<p>
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Linux IA64 / GCC
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</p>
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</td>
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<td>
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<p>
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Peak=0.99 Mean=0.12
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</p>
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<p>
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(<a href="http://www.gnu.org/software/libc/" target="_top">GNU C Lib</a>
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Peak 0)
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</p>
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</td>
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<td>
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<p>
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Pek=1.2 Mean=0.6
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</p>
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<p>
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(<a href="http://www.gnu.org/software/libc/" target="_top">GNU C Lib</a>
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Peak 0)
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</p>
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</td>
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<td>
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<p>
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Peak=0.86 Mean=0.16
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</p>
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<p>
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(<a href="http://www.gnu.org/software/libc/" target="_top">GNU C Lib</a>
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Peak 0)
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</p>
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</td>
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<td>
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<p>
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Peak=2.3 Mean=0.69
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</p>
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<p>
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(<a href="http://www.gnu.org/software/libc/" target="_top">GNU C Lib</a>
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Peak 0)
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</p>
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</td>
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</tr>
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<tr>
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<td>
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<p>
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113
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</p>
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</td>
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<td>
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<p>
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HPUX IA64, aCC A.06.06
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</p>
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</td>
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<td>
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<p>
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Peak=0.96 Mean=0.13
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</p>
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<p>
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(<a href="http://docs.hp.com/en/B9106-90010/index.html" target="_top">HP-UX
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C Library</a> Peak 0)
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</p>
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</td>
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<td>
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<p>
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Peak=0.99 Mean=0.53
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</p>
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<p>
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(<a href="http://docs.hp.com/en/B9106-90010/index.html" target="_top">HP-UX
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C Library</a> Peak 0)
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</p>
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</td>
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<td>
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<p>
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Peak=0.9 Mean=0.4
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</p>
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<p>
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(<a href="http://docs.hp.com/en/B9106-90010/index.html" target="_top">HP-UX
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C Library</a> Peak 0)
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</p>
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</td>
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<td>
|
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<p>
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Peak=3.0 Mean=0.9
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</p>
|
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<p>
|
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(<a href="http://docs.hp.com/en/B9106-90010/index.html" target="_top">HP-UX
|
|
C Library</a> Peak 0)
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</p>
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</td>
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</tr>
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</tbody>
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</table></div>
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<h5>
|
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<a name="math_toolkit.sf_gamma.lgamma.h3"></a>
|
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<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>
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</h5>
|
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<p>
|
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The main tests for this function involve comparisons against the logs of
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the factorials which can be independently calculated to very high accuracy.
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</p>
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<p>
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Random tests in key problem areas are also used.
|
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</p>
|
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<h5>
|
|
<a name="math_toolkit.sf_gamma.lgamma.h4"></a>
|
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<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>
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</h5>
|
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<p>
|
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The generic version of this function is implemented by combining the series
|
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and continued fraction representations for the incomplete gamma function:
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</p>
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<p>
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<span class="inlinemediaobject"><img src="../../../equations/lgamm2.png"></span>
|
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</p>
|
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<p>
|
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where <span class="emphasis"><em>l</em></span> is an arbitrary integration limit: choosing
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<code class="literal">l = max(10, a)</code> seems to work fairly well. For negative
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<span class="emphasis"><em>z</em></span> the logarithm version of the reflection formula is
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used:
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</p>
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<p>
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<span class="inlinemediaobject"><img src="../../../equations/lgamm3.png"></span>
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</p>
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<p>
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For types of known precision, the <a class="link" href="../lanczos.html" title="The Lanczos Approximation">Lanczos
|
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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>
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maps type T to an appropriate approximation. The logarithmic version of the
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<a class="link" href="../lanczos.html" title="The Lanczos Approximation">Lanczos approximation</a> is:
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</p>
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<p>
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<span class="inlinemediaobject"><img src="../../../equations/lgamm4.png"></span>
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</p>
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<p>
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Where L<sub>e,g</sub>   is the Lanczos sum, scaled by e<sup>g</sup>.
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</p>
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<p>
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As before the reflection formula is used for <span class="emphasis"><em>z < 0</em></span>.
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</p>
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<p>
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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
|
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approximation</a> suffers very badly from cancellation error: indeed for
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values sufficiently close to 1 or 2, arbitrarily large relative errors can
|
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be obtained (even though the absolute error is tiny).
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</p>
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<p>
|
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For types with up to 113 bits of precision (up to and including 128-bit long
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doubles), root-preserving rational approximations <a class="link" href="../sf_implementation.html#math_toolkit.sf_implementation.rational_approximations_used">devised
|
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by JM</a> are used over the intervals [1,2] and [2,3]. Over the interval
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[2,3] the approximation form used is:
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</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.png"></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.png"></span>
|
|
</p>
|
|
</div>
|
|
<table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr>
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<td align="left"></td>
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<td align="right"><div class="copyright-footer">Copyright © 2006-2010, 2012, 2013 Paul A. Bristow, Christopher Kormanyos,
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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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</p>
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</div></td>
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</tr></table>
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