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<div class="titlepage"><div><div><h3 class="title">
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<a name="math_toolkit.sf_gamma.tgamma"></a><a class="link" href="tgamma.html" title="Gamma">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.tgamma.h0"></a>
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<span class="phrase"><a name="math_toolkit.sf_gamma.tgamma.synopsis"></a></span><a class="link" href="tgamma.html#math_toolkit.sf_gamma.tgamma.synopsis">Synopsis</a>
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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">tgamma</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 15. 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">tgamma</span><span class="special">(</span><span class="identifier">T</span> <span class="identifier">z</span><span class="special">,</span> <span class="keyword">const</span> <a class="link" href="../../policy.html" title="Chapter 15. 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">tgamma1pm1</span><span class="special">(</span><span class="identifier">T</span> <span class="identifier">dz</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 15. 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">tgamma1pm1</span><span class="special">(</span><span class="identifier">T</span> <span class="identifier">dz</span><span class="special">,</span> <span class="keyword">const</span> <a class="link" href="../../policy.html" title="Chapter 15. 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.tgamma.h1"></a>
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<span class="phrase"><a name="math_toolkit.sf_gamma.tgamma.description"></a></span><a class="link" href="tgamma.html#math_toolkit.sf_gamma.tgamma.description">Description</a>
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</h5>
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<pre class="programlisting"><span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">></span>
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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">tgamma</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 15. 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">tgamma</span><span class="special">(</span><span class="identifier">T</span> <span class="identifier">z</span><span class="special">,</span> <span class="keyword">const</span> <a class="link" href="../../policy.html" title="Chapter 15. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&);</span>
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</pre>
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<p>
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Returns the "true gamma" (hence name tgamma) of value z:
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</p>
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<p>
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<span class="inlinemediaobject"><img src="../../../equations/gamm1.svg"></span>
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</p>
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<p>
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<span class="inlinemediaobject"><img src="../../../graphs/tgamma.svg" align="middle"></span>
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</p>
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<p>
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The final <a class="link" href="../../policy.html" title="Chapter 15. 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 15. 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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There are effectively two versions of the <a href="http://en.wikipedia.org/wiki/Gamma_function" target="_top">tgamma</a>
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function internally: a fully generic version that is slow, but reasonably
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accurate, and a much more efficient approximation that is used where the
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number of digits in the significand 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 this function is computed using the <a class="link" href="../result_type.html" title="Calculation of the Type of the Result"><span class="emphasis"><em>result
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type calculation rules</em></span></a>: the result is <code class="computeroutput"><span class="keyword">double</span></code>
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when T is an integer type, and T otherwise.
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</p>
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<pre class="programlisting"><span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">></span>
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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">tgamma1pm1</span><span class="special">(</span><span class="identifier">T</span> <span class="identifier">dz</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 15. 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">tgamma1pm1</span><span class="special">(</span><span class="identifier">T</span> <span class="identifier">dz</span><span class="special">,</span> <span class="keyword">const</span> <a class="link" href="../../policy.html" title="Chapter 15. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&);</span>
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</pre>
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<p>
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Returns <code class="computeroutput"><span class="identifier">tgamma</span><span class="special">(</span><span class="identifier">dz</span> <span class="special">+</span> <span class="number">1</span><span class="special">)</span> <span class="special">-</span> <span class="number">1</span></code>.
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Internally the implementation does not make use of the addition and subtraction
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implied by the definition, leading to accurate results even for very small
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<code class="computeroutput"><span class="identifier">dz</span></code>. However, the implementation
|
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is capped to either 35 digit accuracy, or to the precision of the <a class="link" href="../lanczos.html" title="The Lanczos Approximation">Lanczos
|
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approximation</a> associated with type T, whichever is more accurate.
|
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</p>
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<p>
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The return type of this function is computed using the <a class="link" href="../result_type.html" title="Calculation of the Type of the Result"><span class="emphasis"><em>result
|
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type calculation rules</em></span></a>: the result is <code class="computeroutput"><span class="keyword">double</span></code>
|
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when T is an integer type, and T otherwise.
|
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</p>
|
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<p>
|
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The final <a class="link" href="../../policy.html" title="Chapter 15. 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 15. 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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<h5>
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<a name="math_toolkit.sf_gamma.tgamma.h2"></a>
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<span class="phrase"><a name="math_toolkit.sf_gamma.tgamma.accuracy"></a></span><a class="link" href="tgamma.html#math_toolkit.sf_gamma.tgamma.accuracy">Accuracy</a>
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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 other common libraries. Unless otherwise specified any floating point
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type that is narrower 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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<div class="table">
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<a name="math_toolkit.sf_gamma.tgamma.table_tgamma"></a><p class="title"><b>Table 6.1. Error rates for tgamma</b></p>
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<div class="table-contents"><table class="table" summary="Error rates for tgamma">
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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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</colgroup>
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<thead><tr>
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<th>
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</th>
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<th>
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<p>
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Microsoft Visual C++ version 12.0<br> Win32<br> double
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</p>
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</th>
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<th>
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<p>
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GNU C++ version 5.1.0<br> linux<br> double
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</p>
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</th>
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<th>
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<p>
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GNU C++ version 5.1.0<br> linux<br> long double
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</p>
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</th>
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<th>
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<p>
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Sun compiler version 0x5130<br> Sun Solaris<br> long double
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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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factorials
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</p>
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</td>
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<td>
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<p>
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<span class="blue">Max = 1.85ε (Mean = 0.491ε)</span><br> <br>
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(<span class="emphasis"><em><math.h>:</em></span> Max = 3.17ε (Mean = 0.928ε))
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</p>
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</td>
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<td>
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<p>
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<span class="blue">Max = 0ε (Mean = 0ε)</span><br> <br> (<span class="emphasis"><em>GSL
|
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1.16:</em></span> Max = 3.95ε (Mean = 0.783ε))<br> (<span class="emphasis"><em>Rmath
|
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3.0.2:</em></span> Max = 314ε (Mean = 93.4ε))<br> (<span class="emphasis"><em>Cephes:</em></span>
|
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Max = 3.19ε (Mean = 0.884ε))
|
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|
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</p>
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</td>
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<td>
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<p>
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<span class="blue">Max = 1.96ε (Mean = 0.483ε)</span><br> <br>
|
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(<span class="emphasis"><em><tr1/cmath>:</em></span> Max = 1.66ε (Mean = 0.584ε))<br>
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(<span class="emphasis"><em><math.h>:</em></span> Max = 1.66ε (Mean = 0.584ε))
|
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</p>
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</td>
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<td>
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<p>
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<span class="blue">Max = 172ε (Mean = 41ε)</span><br> <br>
|
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(<span class="emphasis"><em><math.h>:</em></span> Max = 0ε (Mean = 0ε))
|
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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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near 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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<span class="blue">Max = 1.96ε (Mean = 0.684ε)</span><br> <br>
|
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(<span class="emphasis"><em><math.h>:</em></span> Max = 1ε (Mean = 0.405ε))
|
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</p>
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</td>
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<td>
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<p>
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<span class="blue">Max = 0ε (Mean = 0ε)</span><br> <br> (<span class="emphasis"><em>GSL
|
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1.16:</em></span> Max = 4.51ε (Mean = 1.92ε))<br> (<span class="emphasis"><em>Rmath
|
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3.0.2:</em></span> Max = 1ε (Mean = 0.335ε))<br> (<span class="emphasis"><em>Cephes:</em></span>
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Max = 1ε (Mean = 0.548ε))
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</p>
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</td>
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<td>
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<p>
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<span class="blue">Max = 2ε (Mean = 0.73ε)</span><br> <br>
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(<span class="emphasis"><em><tr1/cmath>:</em></span> Max = 1ε (Mean = 0.376ε))<br>
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(<span class="emphasis"><em><math.h>:</em></span> Max = 1ε (Mean = 0.376ε))
|
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</p>
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</td>
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<td>
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<p>
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<span class="blue">Max = 2ε (Mean = 0.647ε)</span><br> <br>
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(<span class="emphasis"><em><math.h>:</em></span> Max = 0.5ε (Mean = 0.0791ε))
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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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near 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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<span class="blue">Max = 2ε (Mean = 0.865ε)</span><br> <br>
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(<span class="emphasis"><em><math.h>:</em></span> Max = 1ε (Mean = 0.4ε))
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</p>
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|
</td>
|
||
|
|
<td>
|
||
|
|
<p>
|
||
|
|
<span class="blue">Max = 0ε (Mean = 0ε)</span><br> <br> (<span class="emphasis"><em>GSL
|
||
|
|
1.16:</em></span> Max = 4.41ε (Mean = 1.81ε))<br> (<span class="emphasis"><em>Rmath
|
||
|
|
3.0.2:</em></span> Max = 1ε (Mean = 0.32ε))<br> (<span class="emphasis"><em>Cephes:</em></span>
|
||
|
|
Max = 1ε (Mean = 0.518ε))
|
||
|
|
</p>
|
||
|
|
</td>
|
||
|
|
<td>
|
||
|
|
<p>
|
||
|
|
<span class="blue">Max = 2ε (Mean = 0.85ε)</span><br> <br>
|
||
|
|
(<span class="emphasis"><em><tr1/cmath>:</em></span> Max = 0.918ε (Mean = 0.203ε))<br>
|
||
|
|
(<span class="emphasis"><em><math.h>:</em></span> Max = 0.918ε (Mean = 0.203ε))
|
||
|
|
</p>
|
||
|
|
</td>
|
||
|
|
<td>
|
||
|
|
<p>
|
||
|
|
<span class="blue">Max = 3.01ε (Mean = 1.06ε)</span><br> <br>
|
||
|
|
(<span class="emphasis"><em><math.h>:</em></span> Max = 1ε (Mean = 0.175ε))
|
||
|
|
</p>
|
||
|
|
</td>
|
||
|
|
</tr>
|
||
|
|
<tr>
|
||
|
|
<td>
|
||
|
|
<p>
|
||
|
|
near 2
|
||
|
|
</p>
|
||
|
|
</td>
|
||
|
|
<td>
|
||
|
|
<p>
|
||
|
|
<span class="blue">Max = 2ε (Mean = 0.995ε)</span><br> <br>
|
||
|
|
(<span class="emphasis"><em><math.h>:</em></span> Max = 0ε (Mean = 0ε))
|
||
|
|
</p>
|
||
|
|
</td>
|
||
|
|
<td>
|
||
|
|
<p>
|
||
|
|
<span class="blue">Max = 0ε (Mean = 0ε)</span><br> <br> (<span class="emphasis"><em>GSL
|
||
|
|
1.16:</em></span> Max = 7.95ε (Mean = 3.12ε))<br> (<span class="emphasis"><em>Rmath
|
||
|
|
3.0.2:</em></span> Max = 1ε (Mean = 0.191ε))<br> (<span class="emphasis"><em>Cephes:</em></span>
|
||
|
|
Max = 1.09ε (Mean = 0.502ε))
|
||
|
|
</p>
|
||
|
|
</td>
|
||
|
|
<td>
|
||
|
|
<p>
|
||
|
|
<span class="blue">Max = 2ε (Mean = 0.913ε)</span><br> <br>
|
||
|
|
(<span class="emphasis"><em><tr1/cmath>:</em></span> Max = 0.558ε (Mean = 0.298ε))<br>
|
||
|
|
(<span class="emphasis"><em><math.h>:</em></span> Max = 0.558ε (Mean = 0.298ε))
|
||
|
|
</p>
|
||
|
|
</td>
|
||
|
|
<td>
|
||
|
|
<p>
|
||
|
|
<span class="blue">Max = 5.01ε (Mean = 1.89ε)</span><br> <br>
|
||
|
|
(<span class="emphasis"><em><math.h>:</em></span> Max = 0ε (Mean = 0ε))
|
||
|
|
</p>
|
||
|
|
</td>
|
||
|
|
</tr>
|
||
|
|
<tr>
|
||
|
|
<td>
|
||
|
|
<p>
|
||
|
|
near -10
|
||
|
|
</p>
|
||
|
|
</td>
|
||
|
|
<td>
|
||
|
|
<p>
|
||
|
|
<span class="blue">Max = 1.73ε (Mean = 0.729ε)</span><br> <br>
|
||
|
|
(<span class="emphasis"><em><math.h>:</em></span> Max = 0.866ε (Mean = 0.445ε))
|
||
|
|
</p>
|
||
|
|
</td>
|
||
|
|
<td>
|
||
|
|
<p>
|
||
|
|
<span class="blue">Max = 0ε (Mean = 0ε)</span><br> <br> (<span class="emphasis"><em>GSL
|
||
|
|
1.16:</em></span> Max = 2.6ε (Mean = 1.05ε))<br> (<span class="emphasis"><em>Rmath
|
||
|
|
3.0.2:</em></span> Max = 6.34e+05ε (Mean = 1.2e+05ε))<br> (<span class="emphasis"><em>Cephes:</em></span>
|
||
|
|
Max = 2.6ε (Mean = 0.956ε))
|
||
|
|
</p>
|
||
|
|
</td>
|
||
|
|
<td>
|
||
|
|
<p>
|
||
|
|
<span class="blue">Max = 2.6ε (Mean = 0.985ε)</span><br> <br>
|
||
|
|
(<span class="emphasis"><em><tr1/cmath>:</em></span> Max = 2.26ε (Mean = 1.08ε))<br>
|
||
|
|
(<span class="emphasis"><em><math.h>:</em></span> Max = 2.26ε (Mean = 1.08ε))
|
||
|
|
</p>
|
||
|
|
</td>
|
||
|
|
<td>
|
||
|
|
<p>
|
||
|
|
<span class="blue">Max = 1.75ε (Mean = 0.819ε)</span><br> <br>
|
||
|
|
(<span class="emphasis"><em><math.h>:</em></span> Max = 0ε (Mean = 0ε))
|
||
|
|
</p>
|
||
|
|
</td>
|
||
|
|
</tr>
|
||
|
|
<tr>
|
||
|
|
<td>
|
||
|
|
<p>
|
||
|
|
near -55
|
||
|
|
</p>
|
||
|
|
</td>
|
||
|
|
<td>
|
||
|
|
<p>
|
||
|
|
<span class="blue">Max = 1.8ε (Mean = 0.817ε)</span><br> <br>
|
||
|
|
(<span class="emphasis"><em><math.h>:</em></span> Max = 3.87e+004ε (Mean = 6.71e+003ε))
|
||
|
|
</p>
|
||
|
|
</td>
|
||
|
|
<td>
|
||
|
|
<p>
|
||
|
|
<span class="blue">Max = 0ε (Mean = 0ε)</span><br> <br> (<span class="emphasis"><em>GSL
|
||
|
|
1.16:</em></span> Max = 1.8ε (Mean = 0.782ε))<br> (<span class="emphasis"><em>Rmath
|
||
|
|
3.0.2:</em></span> Max = 6.36e+06ε (Mean = 1.13e+06ε))<br> (<span class="emphasis"><em>Cephes:</em></span>
|
||
|
|
Max = 2.7ε (Mean = 0.988ε))
|
||
|
|
</p>
|
||
|
|
</td>
|
||
|
|
<td>
|
||
|
|
<p>
|
||
|
|
<span class="blue">Max = 1.8ε (Mean = 0.847ε)</span><br> <br>
|
||
|
|
(<span class="emphasis"><em><tr1/cmath>:</em></span> Max = 1.79ε (Mean = 0.75ε))<br>
|
||
|
|
(<span class="emphasis"><em><math.h>:</em></span> Max = 1.79ε (Mean = 0.75ε))
|
||
|
|
</p>
|
||
|
|
</td>
|
||
|
|
<td>
|
||
|
|
<p>
|
||
|
|
<span class="blue">Max = 98.5ε (Mean = 53.4ε)</span><br> <br>
|
||
|
|
(<span class="emphasis"><em><math.h>:</em></span> Max = 0ε (Mean = 0ε))
|
||
|
|
</p>
|
||
|
|
</td>
|
||
|
|
</tr>
|
||
|
|
</tbody>
|
||
|
|
</table></div>
|
||
|
|
</div>
|
||
|
|
<br class="table-break"><div class="table">
|
||
|
|
<a name="math_toolkit.sf_gamma.tgamma.table_tgamma1pm1"></a><p class="title"><b>Table 6.2. Error rates for tgamma1pm1</b></p>
|
||
|
|
<div class="table-contents"><table class="table" summary="Error rates for tgamma1pm1">
|
||
|
|
<colgroup>
|
||
|
|
<col>
|
||
|
|
<col>
|
||
|
|
<col>
|
||
|
|
<col>
|
||
|
|
<col>
|
||
|
|
</colgroup>
|
||
|
|
<thead><tr>
|
||
|
|
<th>
|
||
|
|
</th>
|
||
|
|
<th>
|
||
|
|
<p>
|
||
|
|
Microsoft Visual C++ version 12.0<br> Win32<br> double
|
||
|
|
</p>
|
||
|
|
</th>
|
||
|
|
<th>
|
||
|
|
<p>
|
||
|
|
GNU C++ version 5.1.0<br> linux<br> double
|
||
|
|
</p>
|
||
|
|
</th>
|
||
|
|
<th>
|
||
|
|
<p>
|
||
|
|
GNU C++ version 5.1.0<br> linux<br> long double
|
||
|
|
</p>
|
||
|
|
</th>
|
||
|
|
<th>
|
||
|
|
<p>
|
||
|
|
Sun compiler version 0x5130<br> Sun Solaris<br> long double
|
||
|
|
</p>
|
||
|
|
</th>
|
||
|
|
</tr></thead>
|
||
|
|
<tbody><tr>
|
||
|
|
<td>
|
||
|
|
<p>
|
||
|
|
tgamma1pm1(dz)
|
||
|
|
</p>
|
||
|
|
</td>
|
||
|
|
<td>
|
||
|
|
<p>
|
||
|
|
<span class="blue">Max = 0.982ε (Mean = 0.399ε)</span>
|
||
|
|
</p>
|
||
|
|
</td>
|
||
|
|
<td>
|
||
|
|
<p>
|
||
|
|
<span class="blue">Max = 0ε (Mean = 0ε)</span>
|
||
|
|
</p>
|
||
|
|
</td>
|
||
|
|
<td>
|
||
|
|
<p>
|
||
|
|
<span class="blue">Max = 1.12ε (Mean = 0.49ε)</span>
|
||
|
|
</p>
|
||
|
|
</td>
|
||
|
|
<td>
|
||
|
|
<p>
|
||
|
|
<span class="blue">Max = 3.97ε (Mean = 0.713ε)</span>
|
||
|
|
</p>
|
||
|
|
</td>
|
||
|
|
</tr></tbody>
|
||
|
|
</table></div>
|
||
|
|
</div>
|
||
|
|
<br class="table-break"><h5>
|
||
|
|
<a name="math_toolkit.sf_gamma.tgamma.h3"></a>
|
||
|
|
<span class="phrase"><a name="math_toolkit.sf_gamma.tgamma.testing"></a></span><a class="link" href="tgamma.html#math_toolkit.sf_gamma.tgamma.testing">Testing</a>
|
||
|
|
</h5>
|
||
|
|
<p>
|
||
|
|
The gamma is relatively easy to test: factorials and half-integer factorials
|
||
|
|
can be calculated exactly by other means and compared with the gamma function.
|
||
|
|
In addition, some accuracy tests in known tricky areas were computed at high
|
||
|
|
precision using the generic version of this function.
|
||
|
|
</p>
|
||
|
|
<p>
|
||
|
|
The function <code class="computeroutput"><span class="identifier">tgamma1pm1</span></code> is
|
||
|
|
tested against values calculated very naively using the formula <code class="computeroutput"><span class="identifier">tgamma</span><span class="special">(</span><span class="number">1</span><span class="special">+</span><span class="identifier">dz</span><span class="special">)-</span><span class="number">1</span></code> with a
|
||
|
|
lanczos approximation accurate to around 100 decimal digits.
|
||
|
|
</p>
|
||
|
|
<h5>
|
||
|
|
<a name="math_toolkit.sf_gamma.tgamma.h4"></a>
|
||
|
|
<span class="phrase"><a name="math_toolkit.sf_gamma.tgamma.implementation"></a></span><a class="link" href="tgamma.html#math_toolkit.sf_gamma.tgamma.implementation">Implementation</a>
|
||
|
|
</h5>
|
||
|
|
<p>
|
||
|
|
The generic version of the <code class="computeroutput"><span class="identifier">tgamma</span></code>
|
||
|
|
function is implemented Sterling's approximation for lgamma for large z:
|
||
|
|
</p>
|
||
|
|
<p>
|
||
|
|
<span class="inlinemediaobject"><img src="../../../equations/gamma6.svg"></span>
|
||
|
|
</p>
|
||
|
|
<p>
|
||
|
|
Following exponentiation, downward recursion is then used for small values
|
||
|
|
of z.
|
||
|
|
</p>
|
||
|
|
<p>
|
||
|
|
For types of known precision the <a class="link" href="../lanczos.html" title="The Lanczos Approximation">Lanczos
|
||
|
|
approximation</a> is used, a traits class <code class="computeroutput"><span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">::</span><span class="identifier">lanczos</span><span class="special">::</span><span class="identifier">lanczos_traits</span></code>
|
||
|
|
maps type T to an appropriate approximation.
|
||
|
|
</p>
|
||
|
|
<p>
|
||
|
|
For z in the range -20 < z < 1 then recursion is used to shift to z
|
||
|
|
> 1 via:
|
||
|
|
</p>
|
||
|
|
<p>
|
||
|
|
<span class="inlinemediaobject"><img src="../../../equations/gamm3.svg"></span>
|
||
|
|
</p>
|
||
|
|
<p>
|
||
|
|
For very small z, this helps to preserve the identity:
|
||
|
|
</p>
|
||
|
|
<p>
|
||
|
|
<span class="inlinemediaobject"><img src="../../../equations/gamm4.svg"></span>
|
||
|
|
</p>
|
||
|
|
<p>
|
||
|
|
For z < -20 the reflection formula:
|
||
|
|
</p>
|
||
|
|
<p>
|
||
|
|
<span class="inlinemediaobject"><img src="../../../equations/gamm5.svg"></span>
|
||
|
|
</p>
|
||
|
|
<p>
|
||
|
|
is used. Particular care has to be taken to evaluate the <code class="literal">z * sin(π   *
|
||
|
|
z)</code> part: a special routine is used to reduce z prior to multiplying
|
||
|
|
by π   to ensure that the result in is the range [0, π/2]. Without this an excessive
|
||
|
|
amount of error occurs in this region (which is hard enough already, as the
|
||
|
|
rate of change near a negative pole is <span class="emphasis"><em>exceptionally</em></span>
|
||
|
|
high).
|
||
|
|
</p>
|
||
|
|
<p>
|
||
|
|
Finally if the argument is a small integer then table lookup of the factorial
|
||
|
|
is used.
|
||
|
|
</p>
|
||
|
|
<p>
|
||
|
|
The function <code class="computeroutput"><span class="identifier">tgamma1pm1</span></code> is
|
||
|
|
implemented using rational approximations <a class="link" href="../sf_implementation.html#math_toolkit.sf_implementation.rational_approximations_used">devised
|
||
|
|
by JM</a> in the region <code class="computeroutput"><span class="special">-</span><span class="number">0.5</span> <span class="special"><</span> <span class="identifier">dz</span>
|
||
|
|
<span class="special"><</span> <span class="number">2</span></code>.
|
||
|
|
These are the same approximations (and internal routines) that are used for
|
||
|
|
<a class="link" href="lgamma.html" title="Log Gamma">lgamma</a>, and so aren't
|
||
|
|
detailed further here. The result of the approximation is <code class="computeroutput"><span class="identifier">log</span><span class="special">(</span><span class="identifier">tgamma</span><span class="special">(</span><span class="identifier">dz</span><span class="special">+</span><span class="number">1</span><span class="special">))</span></code> which can
|
||
|
|
fed into <a class="link" href="../powers/expm1.html" title="expm1">expm1</a> to give the
|
||
|
|
desired result. Outside the range <code class="computeroutput"><span class="special">-</span><span class="number">0.5</span> <span class="special"><</span> <span class="identifier">dz</span>
|
||
|
|
<span class="special"><</span> <span class="number">2</span></code>
|
||
|
|
then the naive formula <code class="computeroutput"><span class="identifier">tgamma1pm1</span><span class="special">(</span><span class="identifier">dz</span><span class="special">)</span>
|
||
|
|
<span class="special">=</span> <span class="identifier">tgamma</span><span class="special">(</span><span class="identifier">dz</span><span class="special">+</span><span class="number">1</span><span class="special">)-</span><span class="number">1</span></code>
|
||
|
|
can be used directly.
|
||
|
|
</p>
|
||
|
|
</div>
|
||
|
|
<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 © 2006-2010, 2012-2014 Nikhar Agrawal,
|
||
|
|
Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos, Hubert
|
||
|
|
Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Johan Råde, Gautam Sewani,
|
||
|
|
Benjamin Sobotta, Thijs van den Berg, Daryle Walker and Xiaogang Zhang<p>
|
||
|
|
Distributed under the Boost Software License, Version 1.0. (See accompanying
|
||
|
|
file LICENSE_1_0.txt or copy at <a href="http://www.boost.org/LICENSE_1_0.txt" target="_top">http://www.boost.org/LICENSE_1_0.txt</a>)
|
||
|
|
</p>
|
||
|
|
</div></td>
|
||
|
|
</tr></table>
|
||
|
|
<hr>
|
||
|
|
<div class="spirit-nav">
|
||
|
|
<a accesskey="p" href="../sf_gamma.html"><img src="../../../../../../doc/src/images/prev.png" alt="Prev"></a><a accesskey="u" href="../sf_gamma.html"><img src="../../../../../../doc/src/images/up.png" alt="Up"></a><a accesskey="h" href="../../index.html"><img src="../../../../../../doc/src/images/home.png" alt="Home"></a><a accesskey="n" href="lgamma.html"><img src="../../../../../../doc/src/images/next.png" alt="Next"></a>
|
||
|
|
</div>
|
||
|
|
</body>
|
||
|
|
</html>
|