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| <div class="section"> |
| <div class="titlepage"><div><div><h3 class="title"> |
| <a name="math_toolkit.sf_gamma.tgamma"></a><a class="link" href="tgamma.html" title="Gamma">Gamma</a> |
| </h3></div></div></div> |
| <h5> |
| <a name="math_toolkit.sf_gamma.tgamma.h0"></a> |
| <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> |
| </h5> |
| <pre class="programlisting"><span class="preprocessor">#include</span> <span class="special"><</span><span class="identifier">boost</span><span class="special">/</span><span class="identifier">math</span><span class="special">/</span><span class="identifier">special_functions</span><span class="special">/</span><span class="identifier">gamma</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">></span> |
| </pre> |
| <pre class="programlisting"><span class="keyword">namespace</span> <span class="identifier">boost</span><span class="special">{</span> <span class="keyword">namespace</span> <span class="identifier">math</span><span class="special">{</span> |
| |
| <span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">></span> |
| <a class="link" href="../result_type.html" title="Calculation of the Type of the Result"><span class="emphasis"><em>calculated-result-type</em></span></a> <span class="identifier">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="Chapter 14. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">></span> |
| <a class="link" href="../result_type.html" title="Calculation of the Type of the Result"><span class="emphasis"><em>calculated-result-type</em></span></a> <span class="identifier">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 14. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&);</span> |
| |
| <span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">></span> |
| <a class="link" href="../result_type.html" title="Calculation of the Type of the Result"><span class="emphasis"><em>calculated-result-type</em></span></a> <span class="identifier">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="Chapter 14. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">></span> |
| <a class="link" href="../result_type.html" title="Calculation of the Type of the Result"><span class="emphasis"><em>calculated-result-type</em></span></a> <span class="identifier">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 14. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&);</span> |
| |
| <span class="special">}}</span> <span class="comment">// namespaces</span> |
| </pre> |
| <h5> |
| <a name="math_toolkit.sf_gamma.tgamma.h1"></a> |
| <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> |
| </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="../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="Chapter 14. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">></span> |
| <a class="link" href="../result_type.html" title="Calculation of the Type of the Result"><span class="emphasis"><em>calculated-result-type</em></span></a> <span class="identifier">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 14. Policies: Controlling Precision, Error Handling etc">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.svg"></span> |
| </p> |
| <p> |
| <span class="inlinemediaobject"><img src="../../../graphs/tgamma.svg" align="middle"></span> |
| </p> |
| <p> |
| The final <a class="link" href="../../policy.html" title="Chapter 14. Policies: Controlling Precision, Error Handling etc">Policy</a> argument is optional and can |
| be used to control the behaviour of the function: how it handles errors, |
| what level of precision to use etc. Refer to the <a class="link" href="../../policy.html" title="Chapter 14. Policies: Controlling Precision, Error Handling etc">policy |
| documentation for more details</a>. |
| </p> |
| <p> |
| 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="../lanczos.html" title="The Lanczos Approximation">Lanczos |
| approximation</a>. In practice any built in floating point type you will |
| encounter has an appropriate <a class="link" href="../lanczos.html" title="The Lanczos Approximation">Lanczos |
| approximation</a> defined for it. It is also possible, given enough machine |
| time, to generate further <a class="link" href="../lanczos.html" title="The Lanczos Approximation">Lanczos approximation</a>'s |
| using the program libs/math/tools/lanczos_generator.cpp. |
| </p> |
| <p> |
| The return type of 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 |
| 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="../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="Chapter 14. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">></span> |
| <a class="link" href="../result_type.html" title="Calculation of the Type of the Result"><span class="emphasis"><em>calculated-result-type</em></span></a> <span class="identifier">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 14. Policies: Controlling Precision, Error Handling etc">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="../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="../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> |
| The final <a class="link" href="../../policy.html" title="Chapter 14. Policies: Controlling Precision, Error Handling etc">Policy</a> argument is optional and can |
| be used to control the behaviour of the function: how it handles errors, |
| what level of precision to use etc. Refer to the <a class="link" href="../../policy.html" title="Chapter 14. Policies: Controlling Precision, Error Handling etc">policy |
| documentation for more details</a>. |
| </p> |
| <h5> |
| <a name="math_toolkit.sf_gamma.tgamma.h2"></a> |
| <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> |
| </h5> |
| <p> |
| The following table shows the peak errors (in units of epsilon) found on |
| various platforms with various floating point types, along with comparisons |
| to the <a href="http://www.gnu.org/software/gsl/" target="_top">GSL-1.9</a>, <a href="http://www.gnu.org/software/libc/" target="_top">GNU C Lib</a>, <a href="http://docs.hp.com/en/B9106-90010/index.html" target="_top">HP-UX |
| C Library</a> and <a href="http://www.netlib.org/cephes/" target="_top">Cephes</a> |
| libraries. Unless otherwise specified any floating point type that is narrower |
| than the one shown will have <a class="link" href="../relative_error.html#math_toolkit.relative_error.zero_error">effectively |
| zero error</a>. |
| </p> |
| <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> |
| <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> |
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| <td align="right"><div class="copyright-footer">Copyright © 2006-2010, 2012-2014 Nikhar Agrawal, |
| Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos, Hubert |
| Holin, Bruno Lalande, John Maddock, Johan Råde, Gautam Sewani, Benjamin Sobotta, |
| Thijs van den Berg, Daryle Walker and Xiaogang Zhang<p> |
| Distributed under the Boost Software License, Version 1.0. (See accompanying |
| file LICENSE_1_0.txt or copy at <a href="http://www.boost.org/LICENSE_1_0.txt" target="_top">http://www.boost.org/LICENSE_1_0.txt</a>) |
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