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| <div class="section" lang="en"> |
| <div class="titlepage"><div><div><h4 class="title"> |
| <a name="math_toolkit.special.sf_gamma.tgamma"></a><a class="link" href="tgamma.html" title="Gamma"> Gamma</a> |
| </h4></div></div></div> |
| <a name="math_toolkit.special.sf_gamma.tgamma.synopsis"></a><h5> |
| <a name="id1066483"></a> |
| <a class="link" href="tgamma.html#math_toolkit.special.sf_gamma.tgamma.synopsis">Synopsis</a> |
| </h5> |
| <p> |
| |
| </p> |
| <pre class="programlisting"><span class="preprocessor">#include</span> <span class="special"><</span><span class="identifier">boost</span><span class="special">/</span><span class="identifier">math</span><span class="special">/</span><span class="identifier">special_functions</span><span class="special">/</span><span class="identifier">gamma</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">></span> |
| </pre> |
| <p> |
| </p> |
| <pre class="programlisting"><span class="keyword">namespace</span> <span class="identifier">boost</span><span class="special">{</span> <span class="keyword">namespace</span> <span class="identifier">math</span><span class="special">{</span> |
| |
| <span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">></span> |
| <a class="link" href="../../main_overview/result_type.html" title="Calculation of the Type of the Result"><span class="emphasis"><em>calculated-result-type</em></span></a> <span class="identifier">tgamma</span><span class="special">(</span><span class="identifier">T</span> <span class="identifier">z</span><span class="special">);</span> |
| |
| <span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">,</span> <span class="keyword">class</span> <a class="link" href="../../policy.html" title="Policies">Policy</a><span class="special">></span> |
| <a class="link" href="../../main_overview/result_type.html" title="Calculation of the Type of the Result"><span class="emphasis"><em>calculated-result-type</em></span></a> <span class="identifier">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="Policies">Policy</a><span class="special">&);</span> |
| |
| <span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">></span> |
| <a class="link" href="../../main_overview/result_type.html" title="Calculation of the Type of the Result"><span class="emphasis"><em>calculated-result-type</em></span></a> <span class="identifier">tgamma1pm1</span><span class="special">(</span><span class="identifier">T</span> <span class="identifier">dz</span><span class="special">);</span> |
| |
| <span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">,</span> <span class="keyword">class</span> <a class="link" href="../../policy.html" title="Policies">Policy</a><span class="special">></span> |
| <a class="link" href="../../main_overview/result_type.html" title="Calculation of the Type of the Result"><span class="emphasis"><em>calculated-result-type</em></span></a> <span class="identifier">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="Policies">Policy</a><span class="special">&);</span> |
| |
| <span class="special">}}</span> <span class="comment">// namespaces |
| </span></pre> |
| <a name="math_toolkit.special.sf_gamma.tgamma.description"></a><h5> |
| <a name="id1066832"></a> |
| <a class="link" href="tgamma.html#math_toolkit.special.sf_gamma.tgamma.description">Description</a> |
| </h5> |
| <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> |
| <a class="link" href="../../main_overview/result_type.html" title="Calculation of the Type of the Result"><span class="emphasis"><em>calculated-result-type</em></span></a> <span class="identifier">tgamma</span><span class="special">(</span><span class="identifier">T</span> <span class="identifier">z</span><span class="special">);</span> |
| |
| <span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">,</span> <span class="keyword">class</span> <a class="link" href="../../policy.html" title="Policies">Policy</a><span class="special">></span> |
| <a class="link" href="../../main_overview/result_type.html" title="Calculation of the Type of the Result"><span class="emphasis"><em>calculated-result-type</em></span></a> <span class="identifier">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="Policies">Policy</a><span class="special">&);</span> |
| </pre> |
| <p> |
| Returns the "true gamma" (hence name tgamma) of value z: |
| </p> |
| <p> |
| <span class="inlinemediaobject"><img src="../../../../equations/gamm1.png"></span> |
| </p> |
| <p> |
| <span class="inlinemediaobject"><img src="../../../../graphs/tgamma.png" align="middle"></span> |
| </p> |
| <p> |
| </p> |
| <p> |
| The final <a class="link" href="../../policy.html" title="Policies">Policy</a> argument |
| is optional and can be used to control the behaviour of the function: |
| how it handles errors, what level of precision to use etc. Refer to the |
| <a class="link" href="../../policy.html" title="Policies">policy documentation for more details</a>. |
| </p> |
| <p> |
| </p> |
| <p> |
| There are effectively two versions of the <a href="http://en.wikipedia.org/wiki/Gamma_function" target="_top">tgamma</a> |
| function internally: a fully generic version that is slow, but reasonably |
| accurate, and a much more efficient approximation that is used where the |
| number of digits in the significand of T correspond to a certain <a class="link" href="../../backgrounders/lanczos.html" title="The Lanczos Approximation">Lanczos approximation</a>. |
| In practice any built in floating point type you will encounter has an |
| appropriate <a class="link" href="../../backgrounders/lanczos.html" title="The Lanczos Approximation">Lanczos |
| approximation</a> defined for it. It is also possible, given enough |
| machine time, to generate further <a class="link" href="../../backgrounders/lanczos.html" title="The Lanczos Approximation">Lanczos |
| approximation</a>'s using the program libs/math/tools/lanczos_generator.cpp. |
| </p> |
| <p> |
| The return type of this function is computed using the <a class="link" href="../../main_overview/result_type.html" title="Calculation of the Type of the Result"><span class="emphasis"><em>result |
| type calculation rules</em></span></a>: the result is <code class="computeroutput"><span class="keyword">double</span></code> |
| when T is an integer type, and T otherwise. |
| </p> |
| <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> |
| <a class="link" href="../../main_overview/result_type.html" title="Calculation of the Type of the Result"><span class="emphasis"><em>calculated-result-type</em></span></a> <span class="identifier">tgamma1pm1</span><span class="special">(</span><span class="identifier">T</span> <span class="identifier">dz</span><span class="special">);</span> |
| |
| <span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">,</span> <span class="keyword">class</span> <a class="link" href="../../policy.html" title="Policies">Policy</a><span class="special">></span> |
| <a class="link" href="../../main_overview/result_type.html" title="Calculation of the Type of the Result"><span class="emphasis"><em>calculated-result-type</em></span></a> <span class="identifier">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="Policies">Policy</a><span class="special">&);</span> |
| </pre> |
| <p> |
| 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>. Internally the implementation |
| does not make use of the addition and subtraction implied by the definition, |
| leading to accurate results even for very small <code class="computeroutput"><span class="identifier">dz</span></code>. |
| However, the implementation is capped to either 35 digit accuracy, or to |
| the precision of the <a class="link" href="../../backgrounders/lanczos.html" title="The Lanczos Approximation">Lanczos |
| approximation</a> associated with type T, whichever is more accurate. |
| </p> |
| <p> |
| The return type of this function is computed using the <a class="link" href="../../main_overview/result_type.html" title="Calculation of the Type of the Result"><span class="emphasis"><em>result |
| type calculation rules</em></span></a>: the result is <code class="computeroutput"><span class="keyword">double</span></code> |
| when T is an integer type, and T otherwise. |
| </p> |
| <p> |
| </p> |
| <p> |
| The final <a class="link" href="../../policy.html" title="Policies">Policy</a> argument |
| is optional and can be used to control the behaviour of the function: |
| how it handles errors, what level of precision to use etc. Refer to the |
| <a class="link" href="../../policy.html" title="Policies">policy documentation for more details</a>. |
| </p> |
| <p> |
| </p> |
| <a name="math_toolkit.special.sf_gamma.tgamma.accuracy"></a><h5> |
| <a name="id1068285"></a> |
| <a class="link" href="tgamma.html#math_toolkit.special.sf_gamma.tgamma.accuracy">Accuracy</a> |
| </h5> |
| <p> |
| The following table shows the peak errors (in units of epsilon) found on |
| various platforms with various floating point types, along with comparisons |
| to the <a href="http://www.gnu.org/software/gsl/" target="_top">GSL-1.9</a>, |
| <a href="http://www.gnu.org/software/libc/" target="_top">GNU C Lib</a>, <a href="http://docs.hp.com/en/B9106-90010/index.html" target="_top">HP-UX C Library</a> |
| and <a href="http://www.netlib.org/cephes/" target="_top">Cephes</a> libraries. |
| Unless otherwise specified any floating point type that is narrower than |
| the one shown will have <a class="link" href="../../backgrounders/relative_error.html#zero_error">effectively zero error</a>. |
| </p> |
| <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=1.9 Mean=0.7 |
| </p> |
| <p> |
| (GSL=3.9) |
| </p> |
| <p> |
| (<a href="http://www.netlib.org/cephes/" target="_top">Cephes</a>=3.0) |
| </p> |
| </td> |
| <td> |
| <p> |
| Peak=2.0 Mean=1.1 |
| </p> |
| <p> |
| (GSL=4.5) |
| </p> |
| <p> |
| (<a href="http://www.netlib.org/cephes/" target="_top">Cephes</a>=1) |
| </p> |
| </td> |
| <td> |
| <p> |
| Peak=2.0 Mean=1.1 |
| </p> |
| <p> |
| (GSL=7.9) |
| </p> |
| <p> |
| (<a href="http://www.netlib.org/cephes/" target="_top">Cephes</a>=1.0) |
| </p> |
| </td> |
| <td> |
| <p> |
| Peak=2.6 Mean=1.3 |
| </p> |
| <p> |
| (GSL=2.5) |
| </p> |
| <p> |
| (<a href="http://www.netlib.org/cephes/" target="_top">Cephes</a>=2.7) |
| </p> |
| </td> |
| </tr> |
| <tr> |
| <td> |
| <p> |
| 64 |
| </p> |
| </td> |
| <td> |
| <p> |
| Linux IA32 / GCC |
| </p> |
| </td> |
| <td> |
| <p> |
| Peak=300 Mean=49.5 |
| </p> |
| <p> |
| (<a href="http://www.gnu.org/software/libc/" target="_top">GNU C Lib</a> |
| Peak=395 Mean=89) |
| </p> |
| </td> |
| <td> |
| <p> |
| Peak=3.0 Mean=1.4 |
| </p> |
| <p> |
| (<a href="http://www.gnu.org/software/libc/" target="_top">GNU C Lib</a> |
| Peak=11 Mean=3.3) |
| </p> |
| </td> |
| <td> |
| <p> |
| Peak=5.0 Mean=1.8 |
| </p> |
| <p> |
| (<a href="http://www.gnu.org/software/libc/" target="_top">GNU C Lib</a> |
| Peak=0.92 Mean=0.2) |
| </p> |
| </td> |
| <td> |
| <p> |
| Peak=157 Mean=65 |
| </p> |
| <p> |
| (<a href="http://www.gnu.org/software/libc/" target="_top">GNU C Lib</a> |
| Peak=205 Mean=108) |
| </p> |
| </td> |
| </tr> |
| <tr> |
| <td> |
| <p> |
| 64 |
| </p> |
| </td> |
| <td> |
| <p> |
| Linux IA64 / GCC |
| </p> |
| </td> |
| <td> |
| <p> |
| <a href="http://www.gnu.org/software/libc/" target="_top">GNU C Lib</a> |
| Peak 2.8 Mean=0.9 |
| </p> |
| <p> |
| (<a href="http://www.gnu.org/software/libc/" target="_top">GNU C Lib</a> |
| Peak 0.7) |
| </p> |
| </td> |
| <td> |
| <p> |
| Peak=4.8 Mean=1.5 |
| </p> |
| <p> |
| (<a href="http://www.gnu.org/software/libc/" target="_top">GNU C Lib</a> |
| Peak 0) |
| </p> |
| </td> |
| <td> |
| <p> |
| Peak=4.8 Mean=1.5 |
| </p> |
| <p> |
| (<a href="http://www.gnu.org/software/libc/" target="_top">GNU C Lib</a> |
| Peak 0) |
| </p> |
| </td> |
| <td> |
| <p> |
| Peak=5.0 Mean=1.7 (<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=2.5 Mean=1.1 |
| </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=3.5 Mean=1.7 |
| </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=3.5 Mean=1.6 |
| </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=5.2 Mean=1.92 |
| </p> |
| <p> |
| (<a href="http://docs.hp.com/en/B9106-90010/index.html" target="_top">HP-UX |
| C Library</a> Peak 0) |
| </p> |
| </td> |
| </tr> |
| </tbody> |
| </table></div> |
| <a name="math_toolkit.special.sf_gamma.tgamma.testing"></a><h5> |
| <a name="id1068759"></a> |
| <a class="link" href="tgamma.html#math_toolkit.special.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> |
| <a name="math_toolkit.special.sf_gamma.tgamma.implementation"></a><h5> |
| <a name="id1068815"></a> |
| <a class="link" href="tgamma.html#math_toolkit.special.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 by combining the series and continued fraction |
| representations for the incomplete gamma function: |
| </p> |
| <p> |
| <span class="inlinemediaobject"><img src="../../../../equations/gamm2.png"></span> |
| </p> |
| <p> |
| where <span class="emphasis"><em>l</em></span> is an arbitrary integration limit: choosing |
| <code class="literal">l = max(10, a)</code> seems to work fairly well. |
| </p> |
| <p> |
| For types of known precision the <a class="link" href="../../backgrounders/lanczos.html" title="The Lanczos Approximation">Lanczos |
| approximation</a> is used, a traits class <code class="computeroutput"><span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">::</span><span class="identifier">lanczos</span><span class="special">::</span><span class="identifier">lanczos_traits</span></code> |
| maps type T to an appropriate approximation. |
| </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.png"></span> |
| </p> |
| <p> |
| For very small z, this helps to preserve the identity: |
| </p> |
| <p> |
| <span class="inlinemediaobject"><img src="../../../../equations/gamm4.png"></span> |
| </p> |
| <p> |
| For z < -20 the reflection formula: |
| </p> |
| <p> |
| <span class="inlinemediaobject"><img src="../../../../equations/gamm5.png"></span> |
| </p> |
| <p> |
| is used. Particular care has to be taken to evaluate the <code class="computeroutput"><span class="identifier">z</span> <span class="special">*</span> <span class="identifier">sin</span><span class="special">([</span><span class="identifier">pi</span><span class="special">][</span><span class="identifier">space</span><span class="special">]</span> <span class="special">*</span> <span class="identifier">z</span><span class="special">)</span></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="../../backgrounders/implementation.html#math_toolkit.backgrounders.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> |
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| Hubert Holin, Xiaogang Zhang, Bruno Lalande, Johan Råde, Gautam Sewani and |
| Thijs van den Berg<p> |
| Distributed under the Boost Software License, Version 1.0. (See accompanying |
| file LICENSE_1_0.txt or copy at <a href="http://www.boost.org/LICENSE_1_0.txt" target="_top">http://www.boost.org/LICENSE_1_0.txt</a>) |
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