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<div class="section" lang="en">
<div class="titlepage"><div><div><h4 class="title">
<a name="math_toolkit.special.sf_gamma.igamma"></a><a class="link" href="igamma.html" title="Incomplete Gamma Functions"> Incomplete Gamma
Functions</a>
</h4></div></div></div>
<a name="math_toolkit.special.sf_gamma.igamma.synopsis"></a><h5>
<a name="id1080821"></a>
<a class="link" href="igamma.html#math_toolkit.special.sf_gamma.igamma.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">T1</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">T2</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">gamma_p</span><span class="special">(</span><span class="identifier">T1</span> <span class="identifier">a</span><span class="special">,</span> <span class="identifier">T2</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">T1</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">T2</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">gamma_p</span><span class="special">(</span><span class="identifier">T1</span> <span class="identifier">a</span><span class="special">,</span> <span class="identifier">T2</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">T1</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">T2</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">gamma_q</span><span class="special">(</span><span class="identifier">T1</span> <span class="identifier">a</span><span class="special">,</span> <span class="identifier">T2</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">T1</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">T2</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">gamma_q</span><span class="special">(</span><span class="identifier">T1</span> <span class="identifier">a</span><span class="special">,</span> <span class="identifier">T2</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">T1</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">T2</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_lower</span><span class="special">(</span><span class="identifier">T1</span> <span class="identifier">a</span><span class="special">,</span> <span class="identifier">T2</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">T1</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">T2</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_lower</span><span class="special">(</span><span class="identifier">T1</span> <span class="identifier">a</span><span class="special">,</span> <span class="identifier">T2</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">T1</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">T2</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">T1</span> <span class="identifier">a</span><span class="special">,</span> <span class="identifier">T2</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">T1</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">T2</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">T1</span> <span class="identifier">a</span><span class="special">,</span> <span class="identifier">T2</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="special">}}</span> <span class="comment">// namespaces
</span></pre>
<a name="math_toolkit.special.sf_gamma.igamma.description"></a><h5>
<a name="id1082719"></a>
<a class="link" href="igamma.html#math_toolkit.special.sf_gamma.igamma.description">Description</a>
</h5>
<p>
There are four <a href="http://mathworld.wolfram.com/IncompleteGammaFunction.html" target="_top">incomplete
gamma functions</a>: two are normalised versions (also known as <span class="emphasis"><em>regularized</em></span>
incomplete gamma functions) that return values in the range [0, 1], and
two are non-normalised and return values in the range [0, Γ(a)]. Users interested
in statistical applications should use the <a href="http://mathworld.wolfram.com/RegularizedGammaFunction.html" target="_top">normalised
versions (gamma_p and gamma_q)</a>.
</p>
<p>
All of these functions require <span class="emphasis"><em>a > 0</em></span> and <span class="emphasis"><em>z
>= 0</em></span>, otherwise they return the result of <a class="link" href="../../main_overview/error_handling.html#domain_error">domain_error</a>.
</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>
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> when T1 and T2 are different types,
otherwise the return type is simply T1.
</p>
<pre class="programlisting"><span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">T1</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">T2</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">gamma_p</span><span class="special">(</span><span class="identifier">T1</span> <span class="identifier">a</span><span class="special">,</span> <span class="identifier">T2</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">T1</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">T2</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">Policy</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">gamma_p</span><span class="special">(</span><span class="identifier">T1</span> <span class="identifier">a</span><span class="special">,</span> <span class="identifier">T2</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 normalised lower incomplete gamma function of a and z:
</p>
<p>
<span class="inlinemediaobject"><img src="../../../../equations/igamma4.png"></span>
</p>
<p>
This function changes rapidly from 0 to 1 around the point z == a:
</p>
<p>
<span class="inlinemediaobject"><img src="../../../../graphs/gamma_p.png" align="middle"></span>
</p>
<pre class="programlisting"><span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">T1</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">T2</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">gamma_q</span><span class="special">(</span><span class="identifier">T1</span> <span class="identifier">a</span><span class="special">,</span> <span class="identifier">T2</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">T1</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">T2</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">gamma_q</span><span class="special">(</span><span class="identifier">T1</span> <span class="identifier">a</span><span class="special">,</span> <span class="identifier">T2</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 normalised upper incomplete gamma function of a and z:
</p>
<p>
<span class="inlinemediaobject"><img src="../../../../equations/igamma3.png"></span>
</p>
<p>
This function changes rapidly from 1 to 0 around the point z == a:
</p>
<p>
<span class="inlinemediaobject"><img src="../../../../graphs/gamma_q.png" align="middle"></span>
</p>
<pre class="programlisting"><span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">T1</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">T2</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_lower</span><span class="special">(</span><span class="identifier">T1</span> <span class="identifier">a</span><span class="special">,</span> <span class="identifier">T2</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">T1</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">T2</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_lower</span><span class="special">(</span><span class="identifier">T1</span> <span class="identifier">a</span><span class="special">,</span> <span class="identifier">T2</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 full (non-normalised) lower incomplete gamma function of a
and z:
</p>
<p>
<span class="inlinemediaobject"><img src="../../../../equations/igamma2.png"></span>
</p>
<pre class="programlisting"><span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">T1</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">T2</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">T1</span> <span class="identifier">a</span><span class="special">,</span> <span class="identifier">T2</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">T1</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">T2</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">T1</span> <span class="identifier">a</span><span class="special">,</span> <span class="identifier">T2</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 full (non-normalised) upper incomplete gamma function of a
and z:
</p>
<p>
<span class="inlinemediaobject"><img src="../../../../equations/igamma1.png"></span>
</p>
<a name="math_toolkit.special.sf_gamma.igamma.accuracy"></a><h5>
<a name="id1083679"></a>
<a class="link" href="igamma.html#math_toolkit.special.sf_gamma.igamma.accuracy">Accuracy</a>
</h5>
<p>
The following tables give peak and mean relative errors in over various
domains of a and z, along with comparisons to the <a href="http://www.gnu.org/software/gsl/" target="_top">GSL-1.9</a>
and <a href="http://www.netlib.org/cephes/" target="_top">Cephes</a> libraries.
Note that only results for the widest floating point type on the system
are given as narrower types have <a class="link" href="../../backgrounders/relative_error.html#zero_error">effectively
zero error</a>.
</p>
<p>
Note that errors grow as <span class="emphasis"><em>a</em></span> grows larger.
</p>
<p>
Note also that the higher error rates for the 80 and 128 bit long double
results are somewhat misleading: expected results that are zero at 64-bit
double precision may be non-zero - but exceptionally small - with the larger
exponent range of a long double. These results therefore reflect the more
extreme nature of the tests conducted for these types.
</p>
<p>
All values are in units of epsilon.
</p>
<div class="table">
<a name="id1083721"></a><p class="title"><b>Table 18. Errors In the Function gamma_p(a,z)</b></p>
<div class="table-contents"><table class="table" summary="Errors In the Function gamma_p(a,z)">
<colgroup>
<col>
<col>
<col>
<col>
<col>
</colgroup>
<thead><tr>
<th>
<p>
Significand Size
</p>
</th>
<th>
<p>
Platform and Compiler
</p>
</th>
<th>
<p>
0.5 < a < 100
</p>
<p>
and
</p>
<p>
0.01*a < z < 100*a
</p>
</th>
<th>
<p>
1x10<sup>-12</sup> < a < 5x10<sup>-2</sup>
</p>
<p>
and
</p>
<p>
0.01*a < z < 100*a
</p>
</th>
<th>
<p>
1e-6 < a < 1.7x10<sup>6</sup>
</p>
<p>
and
</p>
<p>
1 < z < 100*a
</p>
</th>
</tr></thead>
<tbody>
<tr>
<td>
<p>
53
</p>
</td>
<td>
<p>
Win32, Visual C++ 8
</p>
</td>
<td>
<p>
Peak=36 Mean=9.1
</p>
<p>
(GSL Peak=342 Mean=46)
</p>
<p>
(<a href="http://www.netlib.org/cephes/" target="_top">Cephes</a> Peak=491
Mean=102)
</p>
</td>
<td>
<p>
Peak=4.5 Mean=1.4
</p>
<p>
(GSL Peak=4.8 Mean=0.76)
</p>
<p>
(<a href="http://www.netlib.org/cephes/" target="_top">Cephes</a> Peak=21
Mean=5.6)
</p>
</td>
<td>
<p>
Peak=244 Mean=21
</p>
<p>
(GSL Peak=1022 Mean=1054)
</p>
<p>
(<a href="http://www.netlib.org/cephes/" target="_top">Cephes</a> Peak~8x10<sup>6</sup> Mean~7x10<sup>4</sup>)
</p>
</td>
</tr>
<tr>
<td>
<p>
64
</p>
</td>
<td>
<p>
RedHat Linux IA32, gcc-3.3
</p>
</td>
<td>
<p>
Peak=241 Mean=36
</p>
</td>
<td>
<p>
Peak=4.7 Mean=1.5
</p>
</td>
<td>
<p>
Peak~30,220 Mean=1929
</p>
</td>
</tr>
<tr>
<td>
<p>
64
</p>
</td>
<td>
<p>
Redhat Linux IA64, gcc-3.4
</p>
</td>
<td>
<p>
Peak=41 Mean=10
</p>
</td>
<td>
<p>
Peak=4.7 Mean=1.4
</p>
</td>
<td>
<p>
Peak~30,790 Mean=1864
</p>
</td>
</tr>
<tr>
<td>
<p>
113
</p>
</td>
<td>
<p>
HPUX IA64, aCC A.06.06
</p>
</td>
<td>
<p>
Peak=40.2 Mean=10.2
</p>
</td>
<td>
<p>
Peak=5 Mean=1.6
</p>
</td>
<td>
<p>
Peak=5,476 Mean=440
</p>
</td>
</tr>
</tbody>
</table></div>
</div>
<br class="table-break"><div class="table">
<a name="id1084038"></a><p class="title"><b>Table 19. Errors In the Function gamma_q(a,z)</b></p>
<div class="table-contents"><table class="table" summary="Errors In the Function gamma_q(a,z)">
<colgroup>
<col>
<col>
<col>
<col>
<col>
</colgroup>
<thead><tr>
<th>
<p>
Significand Size
</p>
</th>
<th>
<p>
Platform and Compiler
</p>
</th>
<th>
<p>
0.5 < a < 100
</p>
<p>
and
</p>
<p>
0.01*a < z < 100*a
</p>
</th>
<th>
<p>
1x10<sup>-12</sup> < a < 5x10<sup>-2</sup>
</p>
<p>
and
</p>
<p>
0.01*a < z < 100*a
</p>
</th>
<th>
<p>
1x10<sup>-6</sup> < a < 1.7x10<sup>6</sup>
</p>
<p>
and
</p>
<p>
1 < z < 100*a
</p>
</th>
</tr></thead>
<tbody>
<tr>
<td>
<p>
53
</p>
</td>
<td>
<p>
Win32, Visual C++ 8
</p>
</td>
<td>
<p>
Peak=28.3 Mean=7.2
</p>
<p>
(GSL Peak=201 Mean=13)
</p>
<p>
(<a href="http://www.netlib.org/cephes/" target="_top">Cephes</a> Peak=556
Mean=97)
</p>
</td>
<td>
<p>
Peak=4.8 Mean=1.6
</p>
<p>
(GSL Peak~1.3x10<sup>10</sup> Mean=1x10<sup>+9</sup>)
</p>
<p>
(<a href="http://www.netlib.org/cephes/" target="_top">Cephes</a> Peak~3x10<sup>11</sup> Mean=4x10<sup>10</sup>)
</p>
</td>
<td>
<p>
Peak=469 Mean=33
</p>
<p>
(GSL Peak=27,050 Mean=2159)
</p>
<p>
(<a href="http://www.netlib.org/cephes/" target="_top">Cephes</a> Peak~8x10<sup>6</sup> Mean~7x10<sup>5</sup>)
</p>
</td>
</tr>
<tr>
<td>
<p>
64
</p>
</td>
<td>
<p>
RedHat Linux IA32, gcc-3.3
</p>
</td>
<td>
<p>
Peak=280 Mean=33
</p>
</td>
<td>
<p>
Peak=4.1 Mean=1.6
</p>
</td>
<td>
<p>
Peak=11,490 Mean=732
</p>
</td>
</tr>
<tr>
<td>
<p>
64
</p>
</td>
<td>
<p>
Redhat Linux IA64, gcc-3.4
</p>
</td>
<td>
<p>
Peak=32 Mean=9.4
</p>
</td>
<td>
<p>
Peak=4.7 Mean=1.5
</p>
</td>
<td>
<p>
Peak=6815 Mean=414
</p>
</td>
</tr>
<tr>
<td>
<p>
113
</p>
</td>
<td>
<p>
HPUX IA64, aCC A.06.06
</p>
</td>
<td>
<p>
Peak=37 Mean=10
</p>
</td>
<td>
<p>
Peak=11.2 Mean=2.0
</p>
</td>
<td>
<p>
Peak=4,999 Mean=298
</p>
</td>
</tr>
</tbody>
</table></div>
</div>
<br class="table-break"><div class="table">
<a name="id1084373"></a><p class="title"><b>Table 20. Errors In the Function tgamma_lower(a,z)</b></p>
<div class="table-contents"><table class="table" summary="Errors In the Function tgamma_lower(a,z)">
<colgroup>
<col>
<col>
<col>
<col>
</colgroup>
<thead><tr>
<th>
<p>
Significand Size
</p>
</th>
<th>
<p>
Platform and Compiler
</p>
</th>
<th>
<p>
0.5 < a < 100
</p>
<p>
and
</p>
<p>
0.01*a < z < 100*a
</p>
</th>
<th>
<p>
1x10<sup>-12</sup> < a < 5x10<sup>-2</sup>
</p>
<p>
and
</p>
<p>
0.01*a < z < 100*a
</p>
</th>
</tr></thead>
<tbody>
<tr>
<td>
<p>
53
</p>
</td>
<td>
<p>
Win32, Visual C++ 8
</p>
</td>
<td>
<p>
Peak=5.5 Mean=1.4
</p>
</td>
<td>
<p>
Peak=3.6 Mean=0.78
</p>
</td>
</tr>
<tr>
<td>
<p>
64
</p>
</td>
<td>
<p>
RedHat Linux IA32, gcc-3.3
</p>
</td>
<td>
<p>
Peak=402 Mean=79
</p>
</td>
<td>
<p>
Peak=3.4 Mean=0.8
</p>
</td>
</tr>
<tr>
<td>
<p>
64
</p>
</td>
<td>
<p>
Redhat Linux IA64, gcc-3.4
</p>
</td>
<td>
<p>
Peak=6.8 Mean=1.4
</p>
</td>
<td>
<p>
Peak=3.4 Mean=0.78
</p>
</td>
</tr>
<tr>
<td>
<p>
113
</p>
</td>
<td>
<p>
HPUX IA64, aCC A.06.06
</p>
</td>
<td>
<p>
Peak=6.1 Mean=1.8
</p>
</td>
<td>
<p>
Peak=3.7 Mean=0.89
</p>
</td>
</tr>
</tbody>
</table></div>
</div>
<br class="table-break"><div class="table">
<a name="id1084594"></a><p class="title"><b>Table 21. Errors In the Function tgamma(a,z)</b></p>
<div class="table-contents"><table class="table" summary="Errors In the Function tgamma(a,z)">
<colgroup>
<col>
<col>
<col>
<col>
</colgroup>
<thead><tr>
<th>
<p>
Significand Size
</p>
</th>
<th>
<p>
Platform and Compiler
</p>
</th>
<th>
<p>
0.5 < a < 100
</p>
<p>
and
</p>
<p>
0.01*a < z < 100*a
</p>
</th>
<th>
<p>
1x10<sup>-12</sup> < a < 5x10<sup>-2</sup>
</p>
<p>
and
</p>
<p>
0.01*a < z < 100*a
</p>
</th>
</tr></thead>
<tbody>
<tr>
<td>
<p>
53
</p>
</td>
<td>
<p>
Win32, Visual C++ 8
</p>
</td>
<td>
<p>
Peak=5.9 Mean=1.5
</p>
</td>
<td>
<p>
Peak=1.8 Mean=0.6
</p>
</td>
</tr>
<tr>
<td>
<p>
64
</p>
</td>
<td>
<p>
RedHat Linux IA32, gcc-3.3
</p>
</td>
<td>
<p>
Peak=596 Mean=116
</p>
</td>
<td>
<p>
Peak=3.2 Mean=0.84
</p>
</td>
</tr>
<tr>
<td>
<p>
64
</p>
</td>
<td>
<p>
Redhat Linux IA64, gcc-3.4.4
</p>
</td>
<td>
<p>
Peak=40.2 Mean=2.5
</p>
</td>
<td>
<p>
Peak=3.2 Mean=0.8
</p>
</td>
</tr>
<tr>
<td>
<p>
113
</p>
</td>
<td>
<p>
HPUX IA64, aCC A.06.06
</p>
</td>
<td>
<p>
Peak=364 Mean=17.6
</p>
</td>
<td>
<p>
Peak=12.7 Mean=1.8
</p>
</td>
</tr>
</tbody>
</table></div>
</div>
<br class="table-break"><a name="math_toolkit.special.sf_gamma.igamma.testing"></a><h5>
<a name="id1084826"></a>
<a class="link" href="igamma.html#math_toolkit.special.sf_gamma.igamma.testing">Testing</a>
</h5>
<p>
There are two sets of tests: spot tests compare values taken from <a href="http://functions.wolfram.com/GammaBetaErf/" target="_top">Mathworld's online evaluator</a>
with this implementation to perform a basic "sanity check". Accuracy
tests use data generated at very high precision (using NTL's RR class set
at 1000-bit precision) using this implementation with a very high precision
60-term <a class="link" href="../../backgrounders/lanczos.html" title="The Lanczos Approximation">Lanczos approximation</a>,
and some but not all of the special case handling disabled. This is less
than satisfactory: an independent method should really be used, but apparently
a complete lack of such methods are available. We can't even use a deliberately
naive implementation without special case handling since Legendre's continued
fraction (see below) is unstable for small a and z.
</p>
<a name="math_toolkit.special.sf_gamma.igamma.implementation"></a><h5>
<a name="id1084852"></a>
<a class="link" href="igamma.html#math_toolkit.special.sf_gamma.igamma.implementation">Implementation</a>
</h5>
<p>
These four functions share a common implementation since they are all related
via:
</p>
<p>
1) <span class="inlinemediaobject"><img src="../../../../equations/igamma5.png"></span>
</p>
<p>
2) <span class="inlinemediaobject"><img src="../../../../equations/igamma6.png"></span>
</p>
<p>
3) <span class="inlinemediaobject"><img src="../../../../equations/igamma7.png"></span>
</p>
<p>
The lower incomplete gamma is computed from its series representation:
</p>
<p>
4) <span class="inlinemediaobject"><img src="../../../../equations/igamma8.png"></span>
</p>
<p>
Or by subtraction of the upper integral from either Γ(a) or 1 when <span class="emphasis"><em>x
- (1</em></span>(3x)) > a and x > 1.1/.
</p>
<p>
The upper integral is computed from Legendre's continued fraction representation:
</p>
<p>
5) <span class="inlinemediaobject"><img src="../../../../equations/igamma9.png"></span>
</p>
<p>
When <span class="emphasis"><em>(x > 1.1)</em></span> or by subtraction of the lower integral
from either Γ(a) or 1 when <span class="emphasis"><em>x - (1</em></span>(3x)) < a/.
</p>
<p>
For <span class="emphasis"><em>x < 1.1</em></span> computation of the upper integral is
more complex as the continued fraction representation is unstable in this
area. However there is another series representation for the lower integral:
</p>
<p>
6) <span class="inlinemediaobject"><img src="../../../../equations/igamma10.png"></span>
</p>
<p>
That lends itself to calculation of the upper integral via rearrangement
to:
</p>
<p>
7) <span class="inlinemediaobject"><img src="../../../../equations/igamma11.png"></span>
</p>
<p>
Refer to the documentation for <a class="link" href="../powers/powm1.html" title="powm1">powm1</a>
and <a class="link" href="tgamma.html" title="Gamma">tgamma1pm1</a>
for details of their implementation. Note however that the precision of
<a class="link" href="tgamma.html" title="Gamma">tgamma1pm1</a>
is capped to either around 35 digits, or to that of the <a class="link" href="../../backgrounders/lanczos.html" title="The Lanczos Approximation">Lanczos
approximation</a> associated with type T - if there is one - whichever
of the two is the greater. That therefore imposes a similar limit on the
precision of this function in this region.
</p>
<p>
For <span class="emphasis"><em>x < 1.1</em></span> the crossover point where the result
is ~0.5 no longer occurs for <span class="emphasis"><em>x ~ y</em></span>. Using <span class="emphasis"><em>x
* 0.75 < a</em></span> as the crossover criterion for <span class="emphasis"><em>0.5 <
x <= 1.1</em></span> keeps the maximum value computed (whether it's the
upper or lower interval) to around 0.75. Likewise for <span class="emphasis"><em>x <=
0.5</em></span> then using <span class="emphasis"><em>-0.4 / log(x) < a</em></span> as
the crossover criterion keeps the maximum value computed to around 0.7
(whether it's the upper or lower interval).
</p>
<p>
There are two special cases used when a is an integer or half integer,
and the crossover conditions listed above indicate that we should compute
the upper integral Q. If a is an integer in the range <span class="emphasis"><em>1 <=
a < 30</em></span> then the following finite sum is used:
</p>
<p>
9) <span class="inlinemediaobject"><img src="../../../../equations/igamma1f.png"></span>
</p>
<p>
While for half integers in the range <span class="emphasis"><em>0.5 <= a < 30</em></span>
then the following finite sum is used:
</p>
<p>
10) <span class="inlinemediaobject"><img src="../../../../equations/igamma2f.png"></span>
</p>
<p>
These are both more stable and more efficient than the continued fraction
alternative.
</p>
<p>
When the argument <span class="emphasis"><em>a</em></span> is large, and <span class="emphasis"><em>x ~ a</em></span>
then the series (4) and continued fraction (5) above are very slow to converge.
In this area an expansion due to Temme is used:
</p>
<p>
11) <span class="inlinemediaobject"><img src="../../../../equations/igamma16.png"></span>
</p>
<p>
12) <span class="inlinemediaobject"><img src="../../../../equations/igamma17.png"></span>
</p>
<p>
13) <span class="inlinemediaobject"><img src="../../../../equations/igamma18.png"></span>
</p>
<p>
14) <span class="inlinemediaobject"><img src="../../../../equations/igamma19.png"></span>
</p>
<p>
The double sum is truncated to a fixed number of terms - to give a specific
target precision - and evaluated as a polynomial-of-polynomials. There
are versions for up to 128-bit long double precision: types requiring greater
precision than that do not use these expansions. The coefficients C<sub>k</sub><sup>n</sup> are
computed in advance using the recurrence relations given by Temme. The
zone where these expansions are used is
</p>
<pre class="programlisting"><span class="special">(</span><span class="identifier">a</span> <span class="special">></span> <span class="number">20</span><span class="special">)</span> <span class="special">&&</span> <span class="special">(</span><span class="identifier">a</span> <span class="special"><</span> <span class="number">200</span><span class="special">)</span> <span class="special">&&</span> <span class="identifier">fabs</span><span class="special">(</span><span class="identifier">x</span><span class="special">-</span><span class="identifier">a</span><span class="special">)/</span><span class="identifier">a</span> <span class="special"><</span> <span class="number">0.4</span>
</pre>
<p>
And:
</p>
<pre class="programlisting"><span class="special">(</span><span class="identifier">a</span> <span class="special">></span> <span class="number">200</span><span class="special">)</span> <span class="special">&&</span> <span class="special">(</span><span class="identifier">fabs</span><span class="special">(</span><span class="identifier">x</span><span class="special">-</span><span class="identifier">a</span><span class="special">)/</span><span class="identifier">a</span> <span class="special"><</span> <span class="number">4.5</span><span class="special">/</span><span class="identifier">sqrt</span><span class="special">(</span><span class="identifier">a</span><span class="special">))</span>
</pre>
<p>
The latter range is valid for all types up to 128-bit long doubles, and
is designed to ensure that the result is larger than 10<sup>-6</sup>, the first range
is used only for types up to 80-bit long doubles. These domains are narrower
than the ones recommended by either Temme or Didonato and Morris. However,
using a wider range results in large and inexact (i.e. computed) values
being passed to the <code class="computeroutput"><span class="identifier">exp</span></code>
and <code class="computeroutput"><span class="identifier">erfc</span></code> functions resulting
in significantly larger error rates. In other words there is a fine trade
off here between efficiency and error. The current limits should keep the
number of terms required by (4) and (5) to no more than ~20 at double precision.
</p>
<p>
For the normalised incomplete gamma functions, calculation of the leading
power terms is central to the accuracy of the function. For smallish a
and x combining the power terms with the <a class="link" href="../../backgrounders/lanczos.html" title="The Lanczos Approximation">Lanczos
approximation</a> gives the greatest accuracy:
</p>
<p>
15) <span class="inlinemediaobject"><img src="../../../../equations/igamma12.png"></span>
</p>
<p>
In the event that this causes underflow<span class="emphasis"><em>overflow then the exponent
can be reduced by a factor of /a</em></span> and brought inside the power
term.
</p>
<p>
When a and x are large, we end up with a very large exponent with a base
near one: this will not be computed accurately via the pow function, and
taking logs simply leads to cancellation errors. The worst of the errors
can be avoided by using:
</p>
<p>
16) <span class="inlinemediaobject"><img src="../../../../equations/igamma13.png"></span>
</p>
<p>
when <span class="emphasis"><em>a-x</em></span> is small and a and x are large. There is
still a subtraction and therefore some cancellation errors - but the terms
are small so the absolute error will be small - and it is absolute rather
than relative error that counts in the argument to the <span class="emphasis"><em>exp</em></span>
function. Note that for sufficiently large a and x the errors will still
get you eventually, although this does delay the inevitable much longer
than other methods. Use of <span class="emphasis"><em>log(1+x)-x</em></span> here is inspired
by Temme (see references below).
</p>
<a name="math_toolkit.special.sf_gamma.igamma.references"></a><h5>
<a name="id1085593"></a>
<a class="link" href="igamma.html#math_toolkit.special.sf_gamma.igamma.references">References</a>
</h5>
<div class="itemizedlist"><ul type="disc">
<li>
N. M. Temme, A Set of Algorithms for the Incomplete Gamma Functions,
Probability in the Engineering and Informational Sciences, 8, 1994.
</li>
<li>
N. M. Temme, The Asymptotic Expansion of the Incomplete Gamma Functions,
Siam J. Math Anal. Vol 10 No 4, July 1979, p757.
</li>
<li>
A. R. Didonato and A. H. Morris, Computation of the Incomplete Gamma
Function Ratios and their Inverse. ACM TOMS, Vol 12, No 4, Dec 1986,
p377.
</li>
<li>
W. Gautschi, The Incomplete Gamma Functions Since Tricomi, In Tricomi's
Ideas and Contemporary Applied Mathematics, Atti dei Convegni Lincei,
n. 147, Accademia Nazionale dei Lincei, Roma, 1998, pp. 203--237.
<a href="http://citeseer.ist.psu.edu/gautschi98incomplete.html" target="_top">http://citeseer.ist.psu.edu/gautschi98incomplete.html</a>
</li>
</ul></div>
</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 , 2007, 2008, 2009 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>)
</p>
</div></td>
</tr></table>
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