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| <div class="titlepage"><div><div><h4 class="title"> |
| <a name="math_toolkit.special.sf_gamma.lgamma"></a><a class="link" href="lgamma.html" title="Log Gamma"> Log Gamma</a> |
| </h4></div></div></div> |
| <a name="math_toolkit.special.sf_gamma.lgamma.synopsis"></a><h5> |
| <a name="id1069234"></a> |
| <a class="link" href="lgamma.html#math_toolkit.special.sf_gamma.lgamma.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">lgamma</span><span class="special">(</span><span class="identifier">T</span> <span class="identifier">z</span><span class="special">);</span> |
| |
| <span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">,</span> <span class="keyword">class</span> <a class="link" href="../../policy.html" title="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">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="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">lgamma</span><span class="special">(</span><span class="identifier">T</span> <span class="identifier">z</span><span class="special">,</span> <span class="keyword">int</span><span class="special">*</span> <span class="identifier">sign</span><span class="special">);</span> |
| |
| <span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">,</span> <span class="keyword">class</span> <a class="link" href="../../policy.html" title="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">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="Policies">Policy</a><span class="special">&);</span> |
| |
| <span class="special">}}</span> <span class="comment">// namespaces |
| </span></pre> |
| <a name="math_toolkit.special.sf_gamma.lgamma.description"></a><h5> |
| <a name="id1069617"></a> |
| <a class="link" href="lgamma.html#math_toolkit.special.sf_gamma.lgamma.description">Description</a> |
| </h5> |
| <p> |
| The <a href="http://en.wikipedia.org/wiki/Gamma_function" target="_top">lgamma function</a> |
| is defined by: |
| </p> |
| <p> |
| <span class="inlinemediaobject"><img src="../../../../equations/lgamm1.png"></span> |
| </p> |
| <p> |
| The second form of the function takes a pointer to an integer, which if |
| non-null is set on output to the sign of tgamma(z). |
| </p> |
| <p> |
| </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> |
| <span class="inlinemediaobject"><img src="../../../../graphs/lgamma.png" align="middle"></span> |
| </p> |
| <p> |
| There are effectively two versions of this function internally: a fully |
| generic version that is slow, but reasonably accurate, and a much more |
| efficient approximation that is used where the number of digits in the |
| significand of T correspond to a certain <a class="link" href="../../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 these functions 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 of type <code class="computeroutput"><span class="keyword">double</span></code> if T is an integer type, or type |
| T otherwise. |
| </p> |
| <a name="math_toolkit.special.sf_gamma.lgamma.accuracy"></a><h5> |
| <a name="id1069744"></a> |
| <a class="link" href="lgamma.html#math_toolkit.special.sf_gamma.lgamma.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> |
| <p> |
| Note that while the relative errors near the positive roots of lgamma are |
| very low, the lgamma function has an infinite number of irrational roots |
| for negative arguments: very close to these negative roots only a low absolute |
| error can be guaranteed. |
| </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=0.88 Mean=0.14 |
| </p> |
| <p> |
| (GSL=33) (<a href="http://www.netlib.org/cephes/" target="_top">Cephes</a>=1.5) |
| </p> |
| </td> |
| <td> |
| <p> |
| Peak=0.96 Mean=0.46 |
| </p> |
| <p> |
| (GSL=5.2) (<a href="http://www.netlib.org/cephes/" target="_top">Cephes</a>=1.1) |
| </p> |
| </td> |
| <td> |
| <p> |
| Peak=0.86 Mean=0.46 |
| </p> |
| <p> |
| (GSL=1168) (<a href="http://www.netlib.org/cephes/" target="_top">Cephes</a>~500000) |
| </p> |
| </td> |
| <td> |
| <p> |
| Peak=4.2 Mean=1.3 |
| </p> |
| <p> |
| (GSL=25) (<a href="http://www.netlib.org/cephes/" target="_top">Cephes</a>=1.6) |
| </p> |
| </td> |
| </tr> |
| <tr> |
| <td> |
| <p> |
| 64 |
| </p> |
| </td> |
| <td> |
| <p> |
| Linux IA32 / GCC |
| </p> |
| </td> |
| <td> |
| <p> |
| Peak=1.9 Mean=0.43 |
| </p> |
| <p> |
| (<a href="http://www.gnu.org/software/libc/" target="_top">GNU C Lib</a> |
| Peak=1.7 Mean=0.49) |
| </p> |
| </td> |
| <td> |
| <p> |
| Peak=1.4 Mean=0.57 |
| </p> |
| <p> |
| (<a href="http://www.gnu.org/software/libc/" target="_top">GNU C Lib</a> |
| Peak= 0.96 Mean=0.54) |
| </p> |
| </td> |
| <td> |
| <p> |
| Peak=0.86 Mean=0.35 |
| </p> |
| <p> |
| (<a href="http://www.gnu.org/software/libc/" target="_top">GNU C Lib</a> |
| Peak=0.74 Mean=0.26) |
| </p> |
| </td> |
| <td> |
| <p> |
| Peak=6.0 Mean=1.8 |
| </p> |
| <p> |
| (<a href="http://www.gnu.org/software/libc/" target="_top">GNU C Lib</a> |
| Peak=3.0 Mean=0.86) |
| </p> |
| </td> |
| </tr> |
| <tr> |
| <td> |
| <p> |
| 64 |
| </p> |
| </td> |
| <td> |
| <p> |
| Linux IA64 / GCC |
| </p> |
| </td> |
| <td> |
| <p> |
| Peak=0.99 Mean=0.12 |
| </p> |
| <p> |
| (<a href="http://www.gnu.org/software/libc/" target="_top">GNU C Lib</a> |
| Peak 0) |
| </p> |
| </td> |
| <td> |
| <p> |
| Pek=1.2 Mean=0.6 |
| </p> |
| <p> |
| (<a href="http://www.gnu.org/software/libc/" target="_top">GNU C Lib</a> |
| Peak 0) |
| </p> |
| </td> |
| <td> |
| <p> |
| Peak=0.86 Mean=0.16 |
| </p> |
| <p> |
| (<a href="http://www.gnu.org/software/libc/" target="_top">GNU C Lib</a> |
| Peak 0) |
| </p> |
| </td> |
| <td> |
| <p> |
| Peak=2.3 Mean=0.69 |
| </p> |
| <p> |
| (<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=0.96 Mean=0.13 |
| </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=0.99 Mean=0.53 |
| </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=0.9 Mean=0.4 |
| </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=3.0 Mean=0.9 |
| </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.lgamma.testing"></a><h5> |
| <a name="id1070207"></a> |
| <a class="link" href="lgamma.html#math_toolkit.special.sf_gamma.lgamma.testing">Testing</a> |
| </h5> |
| <p> |
| The main tests for this function involve comparisons against the logs of |
| the factorials which can be independently calculated to very high accuracy. |
| </p> |
| <p> |
| Random tests in key problem areas are also used. |
| </p> |
| <a name="math_toolkit.special.sf_gamma.lgamma.implementation"></a><h5> |
| <a name="id1070228"></a> |
| <a class="link" href="lgamma.html#math_toolkit.special.sf_gamma.lgamma.implementation">Implementation</a> |
| </h5> |
| <p> |
| The generic version of this function is implemented by combining the series |
| and continued fraction representations for the incomplete gamma function: |
| </p> |
| <p> |
| <span class="inlinemediaobject"><img src="../../../../equations/lgamm2.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. For negative |
| <span class="emphasis"><em>z</em></span> the logarithm version of the reflection formula |
| is used: |
| </p> |
| <p> |
| <span class="inlinemediaobject"><img src="../../../../equations/lgamm3.png"></span> |
| </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. The logarithmic version of |
| the <a class="link" href="../../backgrounders/lanczos.html" title="The Lanczos Approximation">Lanczos approximation</a> |
| is: |
| </p> |
| <p> |
| <span class="inlinemediaobject"><img src="../../../../equations/lgamm4.png"></span> |
| </p> |
| <p> |
| Where L<sub>e,g</sub> ​ is the Lanczos sum, scaled by e<sup>g</sup>. |
| </p> |
| <p> |
| As before the reflection formula is used for <span class="emphasis"><em>z < 0</em></span>. |
| </p> |
| <p> |
| When z is very near 1 or 2, then the logarithmic version of the <a class="link" href="../../backgrounders/lanczos.html" title="The Lanczos Approximation">Lanczos |
| approximation</a> suffers very badly from cancellation error: indeed |
| for values sufficiently close to 1 or 2, arbitrarily large relative errors |
| can be obtained (even though the absolute error is tiny). |
| </p> |
| <p> |
| For types with up to 113 bits of precision (up to and including 128-bit |
| long doubles), root-preserving rational approximations <a class="link" href="../../backgrounders/implementation.html#math_toolkit.backgrounders.implementation.rational_approximations_used">devised |
| by JM</a> are used over the intervals [1,2] and [2,3]. Over the interval |
| [2,3] the approximation form used is: |
| </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="../../backgrounders/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="../../backgrounders/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="../../backgrounders/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> |
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| <td align="right"><div class="copyright-footer">Copyright © 2006 , 2007, 2008, 2009, 2010 John Maddock, Paul A. Bristow, |
| 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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