1<html> 2<head> 3<meta http-equiv="Content-Type" content="text/html; charset=UTF-8"> 4<title>Incomplete Gamma Functions</title> 5<link rel="stylesheet" href="../../math.css" type="text/css"> 6<meta name="generator" content="DocBook XSL Stylesheets V1.79.1"> 7<link rel="home" href="../../index.html" title="Math Toolkit 2.12.0"> 8<link rel="up" href="../sf_gamma.html" title="Gamma Functions"> 9<link rel="prev" href="gamma_ratios.html" title="Ratios of Gamma Functions"> 10<link rel="next" href="igamma_inv.html" title="Incomplete Gamma Function Inverses"> 11</head> 12<body bgcolor="white" text="black" link="#0000FF" vlink="#840084" alink="#0000FF"> 13<table cellpadding="2" width="100%"><tr> 14<td valign="top"><img alt="Boost C++ Libraries" width="277" height="86" src="../../../../../../boost.png"></td> 15<td align="center"><a href="../../../../../../index.html">Home</a></td> 16<td align="center"><a href="../../../../../../libs/libraries.htm">Libraries</a></td> 17<td align="center"><a href="http://www.boost.org/users/people.html">People</a></td> 18<td align="center"><a href="http://www.boost.org/users/faq.html">FAQ</a></td> 19<td align="center"><a href="../../../../../../more/index.htm">More</a></td> 20</tr></table> 21<hr> 22<div class="spirit-nav"> 23<a accesskey="p" href="gamma_ratios.html"><img src="../../../../../../doc/src/images/prev.png" alt="Prev"></a><a accesskey="u" href="../sf_gamma.html"><img src="../../../../../../doc/src/images/up.png" alt="Up"></a><a accesskey="h" href="../../index.html"><img src="../../../../../../doc/src/images/home.png" alt="Home"></a><a accesskey="n" href="igamma_inv.html"><img src="../../../../../../doc/src/images/next.png" alt="Next"></a> 24</div> 25<div class="section"> 26<div class="titlepage"><div><div><h3 class="title"> 27<a name="math_toolkit.sf_gamma.igamma"></a><a class="link" href="igamma.html" title="Incomplete Gamma Functions">Incomplete Gamma Functions</a> 28</h3></div></div></div> 29<h5> 30<a name="math_toolkit.sf_gamma.igamma.h0"></a> 31 <span class="phrase"><a name="math_toolkit.sf_gamma.igamma.synopsis"></a></span><a class="link" href="igamma.html#math_toolkit.sf_gamma.igamma.synopsis">Synopsis</a> 32 </h5> 33<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> 34</pre> 35<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> 36 37<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> 38<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">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> 39 40<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="Chapter 21. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">></span> 41<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">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="Chapter 21. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&);</span> 42 43<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> 44<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">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> 45 46<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="Chapter 21. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">></span> 47<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">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="Chapter 21. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&);</span> 48 49<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> 50<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_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> 51 52<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="Chapter 21. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">></span> 53<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_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="Chapter 21. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&);</span> 54 55<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> 56<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">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> 57 58<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="Chapter 21. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">></span> 59<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">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="Chapter 21. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&);</span> 60 61<span class="special">}}</span> <span class="comment">// namespaces</span> 62</pre> 63<h5> 64<a name="math_toolkit.sf_gamma.igamma.h1"></a> 65 <span class="phrase"><a name="math_toolkit.sf_gamma.igamma.description"></a></span><a class="link" href="igamma.html#math_toolkit.sf_gamma.igamma.description">Description</a> 66 </h5> 67<p> 68 There are four <a href="http://mathworld.wolfram.com/IncompleteGammaFunction.html" target="_top">incomplete 69 gamma functions</a>: two are normalised versions (also known as <span class="emphasis"><em>regularized</em></span> 70 incomplete gamma functions) that return values in the range [0, 1], and two 71 are non-normalised and return values in the range [0, Γ(a)]. Users interested 72 in statistical applications should use the <a href="http://mathworld.wolfram.com/RegularizedGammaFunction.html" target="_top">normalised 73 versions (<code class="computeroutput"><span class="identifier">gamma_p</span></code> and <code class="computeroutput"><span class="identifier">gamma_q</span></code>)</a>. 74 </p> 75<p> 76 All of these functions require <span class="emphasis"><em>a > 0</em></span> and <span class="emphasis"><em>z 77 >= 0</em></span>, otherwise they return the result of <a class="link" href="../error_handling.html#math_toolkit.error_handling.domain_error">domain_error</a>. 78 </p> 79<p> 80 The final <a class="link" href="../../policy.html" title="Chapter 21. Policies: Controlling Precision, Error Handling etc">Policy</a> argument is optional and can 81 be used to control the behaviour of the function: how it handles errors, 82 what level of precision to use etc. Refer to the <a class="link" href="../../policy.html" title="Chapter 21. Policies: Controlling Precision, Error Handling etc">policy 83 documentation for more details</a>. 84 </p> 85<p> 86 The return type of these functions is computed using the <a class="link" href="../result_type.html" title="Calculation of the Type of the Result"><span class="emphasis"><em>result 87 type calculation rules</em></span></a> when T1 and T2 are different types, 88 otherwise the return type is simply T1. 89 </p> 90<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> 91<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">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> 92 93<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> 94<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">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="Chapter 21. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&);</span> 95</pre> 96<p> 97 Returns the normalised lower incomplete gamma function of a and z: 98 </p> 99<div class="blockquote"><blockquote class="blockquote"><p> 100 <span class="inlinemediaobject"><img src="../../../equations/igamma4.svg"></span> 101 102 </p></blockquote></div> 103<p> 104 This function changes rapidly from 0 to 1 around the point z == a: 105 </p> 106<div class="blockquote"><blockquote class="blockquote"><p> 107 <span class="inlinemediaobject"><img src="../../../graphs/gamma_p.svg" align="middle"></span> 108 109 </p></blockquote></div> 110<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> 111<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">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> 112 113<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="Chapter 21. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">></span> 114<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">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="Chapter 21. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&);</span> 115</pre> 116<p> 117 Returns the normalised upper incomplete gamma function of a and z: 118 </p> 119<div class="blockquote"><blockquote class="blockquote"><p> 120 <span class="inlinemediaobject"><img src="../../../equations/igamma3.svg"></span> 121 122 </p></blockquote></div> 123<p> 124 This function changes rapidly from 1 to 0 around the point z == a: 125 </p> 126<div class="blockquote"><blockquote class="blockquote"><p> 127 <span class="inlinemediaobject"><img src="../../../graphs/gamma_q.svg" align="middle"></span> 128 129 </p></blockquote></div> 130<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> 131<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_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> 132 133<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="Chapter 21. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">></span> 134<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_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="Chapter 21. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&);</span> 135</pre> 136<p> 137 Returns the full (non-normalised) lower incomplete gamma function of a and 138 z: 139 </p> 140<div class="blockquote"><blockquote class="blockquote"><p> 141 <span class="inlinemediaobject"><img src="../../../equations/igamma2.svg"></span> 142 143 </p></blockquote></div> 144<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> 145<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">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> 146 147<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="Chapter 21. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">></span> 148<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">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="Chapter 21. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&);</span> 149</pre> 150<p> 151 Returns the full (non-normalised) upper incomplete gamma function of a and 152 z: 153 </p> 154<div class="blockquote"><blockquote class="blockquote"><p> 155 <span class="inlinemediaobject"><img src="../../../equations/igamma1.svg"></span> 156 157 </p></blockquote></div> 158<h5> 159<a name="math_toolkit.sf_gamma.igamma.h2"></a> 160 <span class="phrase"><a name="math_toolkit.sf_gamma.igamma.accuracy"></a></span><a class="link" href="igamma.html#math_toolkit.sf_gamma.igamma.accuracy">Accuracy</a> 161 </h5> 162<p> 163 The following tables give peak and mean relative errors in over various domains 164 of a and z, along with comparisons to the <a href="http://www.gnu.org/software/gsl/" target="_top">GSL-1.9</a> 165 and <a href="http://www.netlib.org/cephes/" target="_top">Cephes</a> libraries. 166 Note that only results for the widest floating-point type on the system are 167 given as narrower types have <a class="link" href="../relative_error.html#math_toolkit.relative_error.zero_error">effectively 168 zero error</a>. 169 </p> 170<p> 171 Note that errors grow as <span class="emphasis"><em>a</em></span> grows larger. 172 </p> 173<p> 174 Note also that the higher error rates for the 80 and 128 bit long double 175 results are somewhat misleading: expected results that are zero at 64-bit 176 double precision may be non-zero - but exceptionally small - with the larger 177 exponent range of a long double. These results therefore reflect the more 178 extreme nature of the tests conducted for these types. 179 </p> 180<p> 181 All values are in units of epsilon. 182 </p> 183<div class="table"> 184<a name="math_toolkit.sf_gamma.igamma.table_gamma_p"></a><p class="title"><b>Table 8.9. Error rates for gamma_p</b></p> 185<div class="table-contents"><table class="table" summary="Error rates for gamma_p"> 186<colgroup> 187<col> 188<col> 189<col> 190<col> 191<col> 192</colgroup> 193<thead><tr> 194<th> 195 </th> 196<th> 197 <p> 198 GNU C++ version 7.1.0<br> linux<br> double 199 </p> 200 </th> 201<th> 202 <p> 203 GNU C++ version 7.1.0<br> linux<br> long double 204 </p> 205 </th> 206<th> 207 <p> 208 Sun compiler version 0x5150<br> Sun Solaris<br> long double 209 </p> 210 </th> 211<th> 212 <p> 213 Microsoft Visual C++ version 14.1<br> Win32<br> double 214 </p> 215 </th> 216</tr></thead> 217<tbody> 218<tr> 219<td> 220 <p> 221 tgamma(a, z) medium values 222 </p> 223 </td> 224<td> 225 <p> 226 <span class="blue">Max = 0.955ε (Mean = 0.05ε)</span><br> <br> 227 (<span class="emphasis"><em>GSL 2.1:</em></span> Max = 342ε (Mean = 45.8ε))<br> (<span class="emphasis"><em>Rmath 228 3.2.3:</em></span> Max = 389ε (Mean = 44ε)) 229 </p> 230 </td> 231<td> 232 <p> 233 <span class="blue">Max = 41.6ε (Mean = 8.09ε)</span> 234 </p> 235 </td> 236<td> 237 <p> 238 <span class="blue">Max = 239ε (Mean = 30.2ε)</span> 239 </p> 240 </td> 241<td> 242 <p> 243 <span class="blue">Max = 35.1ε (Mean = 6.98ε)</span> 244 </p> 245 </td> 246</tr> 247<tr> 248<td> 249 <p> 250 tgamma(a, z) small values 251 </p> 252 </td> 253<td> 254 <p> 255 <span class="blue">Max = 0ε (Mean = 0ε)</span><br> <br> (<span class="emphasis"><em>GSL 256 2.1:</em></span> Max = 4.82ε (Mean = 0.758ε))<br> (<span class="emphasis"><em>Rmath 257 3.2.3:</em></span> Max = 1.01ε (Mean = 0.306ε)) 258 </p> 259 </td> 260<td> 261 <p> 262 <span class="blue">Max = 2ε (Mean = 0.464ε)</span> 263 </p> 264 </td> 265<td> 266 <p> 267 <span class="blue">Max = 2ε (Mean = 0.461ε)</span> 268 </p> 269 </td> 270<td> 271 <p> 272 <span class="blue">Max = 1.54ε (Mean = 0.439ε)</span> 273 </p> 274 </td> 275</tr> 276<tr> 277<td> 278 <p> 279 tgamma(a, z) large values 280 </p> 281 </td> 282<td> 283 <p> 284 <span class="blue">Max = 0ε (Mean = 0ε)</span><br> <br> (<span class="emphasis"><em>GSL 285 2.1:</em></span> Max = 1.02e+03ε (Mean = 105ε))<br> (<span class="emphasis"><em>Rmath 286 3.2.3:</em></span> Max = 1.11e+03ε (Mean = 67.5ε)) 287 </p> 288 </td> 289<td> 290 <p> 291 <span class="blue">Max = 3.08e+04ε (Mean = 1.86e+03ε)</span> 292 </p> 293 </td> 294<td> 295 <p> 296 <span class="blue">Max = 3.02e+04ε (Mean = 1.91e+03ε)</span> 297 </p> 298 </td> 299<td> 300 <p> 301 <span class="blue">Max = 243ε (Mean = 20.2ε)</span> 302 </p> 303 </td> 304</tr> 305<tr> 306<td> 307 <p> 308 tgamma(a, z) integer and half integer values 309 </p> 310 </td> 311<td> 312 <p> 313 <span class="blue">Max = 0ε (Mean = 0ε)</span><br> <br> (<span class="emphasis"><em>GSL 314 2.1:</em></span> Max = 128ε (Mean = 22.6ε))<br> (<span class="emphasis"><em>Rmath 315 3.2.3:</em></span> Max = 66.2ε (Mean = 12.2ε)) 316 </p> 317 </td> 318<td> 319 <p> 320 <span class="blue">Max = 11.8ε (Mean = 2.66ε)</span> 321 </p> 322 </td> 323<td> 324 <p> 325 <span class="blue">Max = 71.6ε (Mean = 9.47ε)</span> 326 </p> 327 </td> 328<td> 329 <p> 330 <span class="blue">Max = 13ε (Mean = 2.97ε)</span> 331 </p> 332 </td> 333</tr> 334</tbody> 335</table></div> 336</div> 337<br class="table-break"><div class="table"> 338<a name="math_toolkit.sf_gamma.igamma.table_gamma_q"></a><p class="title"><b>Table 8.10. Error rates for gamma_q</b></p> 339<div class="table-contents"><table class="table" summary="Error rates for gamma_q"> 340<colgroup> 341<col> 342<col> 343<col> 344<col> 345<col> 346</colgroup> 347<thead><tr> 348<th> 349 </th> 350<th> 351 <p> 352 GNU C++ version 7.1.0<br> linux<br> double 353 </p> 354 </th> 355<th> 356 <p> 357 GNU C++ version 7.1.0<br> linux<br> long double 358 </p> 359 </th> 360<th> 361 <p> 362 Sun compiler version 0x5150<br> Sun Solaris<br> long double 363 </p> 364 </th> 365<th> 366 <p> 367 Microsoft Visual C++ version 14.1<br> Win32<br> double 368 </p> 369 </th> 370</tr></thead> 371<tbody> 372<tr> 373<td> 374 <p> 375 tgamma(a, z) medium values 376 </p> 377 </td> 378<td> 379 <p> 380 <span class="blue">Max = 0.927ε (Mean = 0.035ε)</span><br> <br> 381 (<span class="emphasis"><em>GSL 2.1:</em></span> Max = 201ε (Mean = 13.5ε))<br> (<span class="emphasis"><em>Rmath 382 3.2.3:</em></span> Max = 131ε (Mean = 12.7ε)) 383 </p> 384 </td> 385<td> 386 <p> 387 <span class="blue">Max = 32.3ε (Mean = 6.61ε)</span> 388 </p> 389 </td> 390<td> 391 <p> 392 <span class="blue">Max = 199ε (Mean = 26.6ε)</span> 393 </p> 394 </td> 395<td> 396 <p> 397 <span class="blue">Max = 23.7ε (Mean = 4ε)</span> 398 </p> 399 </td> 400</tr> 401<tr> 402<td> 403 <p> 404 tgamma(a, z) small values 405 </p> 406 </td> 407<td> 408 <p> 409 <span class="blue">Max = 0ε (Mean = 0ε)</span><br> <br> (<span class="emphasis"><em>GSL 410 2.1:</em></span> <span class="red">Max = 1.38e+10ε (Mean = 1.05e+09ε))</span><br> 411 (<span class="emphasis"><em>Rmath 3.2.3:</em></span> Max = 65.6ε (Mean = 11ε)) 412 </p> 413 </td> 414<td> 415 <p> 416 <span class="blue">Max = 2.45ε (Mean = 0.885ε)</span> 417 </p> 418 </td> 419<td> 420 <p> 421 <span class="blue">Max = 2.45ε (Mean = 0.819ε)</span> 422 </p> 423 </td> 424<td> 425 <p> 426 <span class="blue">Max = 2.26ε (Mean = 0.74ε)</span> 427 </p> 428 </td> 429</tr> 430<tr> 431<td> 432 <p> 433 tgamma(a, z) large values 434 </p> 435 </td> 436<td> 437 <p> 438 <span class="blue">Max = 0ε (Mean = 0ε)</span><br> <br> (<span class="emphasis"><em>GSL 439 2.1:</em></span> Max = 2.71e+04ε (Mean = 2.16e+03ε))<br> (<span class="emphasis"><em>Rmath 440 3.2.3:</em></span> Max = 1.02e+03ε (Mean = 62.7ε)) 441 </p> 442 </td> 443<td> 444 <p> 445 <span class="blue">Max = 6.82e+03ε (Mean = 414ε)</span> 446 </p> 447 </td> 448<td> 449 <p> 450 <span class="blue">Max = 1.15e+04ε (Mean = 733ε)</span> 451 </p> 452 </td> 453<td> 454 <p> 455 <span class="blue">Max = 469ε (Mean = 31.5ε)</span> 456 </p> 457 </td> 458</tr> 459<tr> 460<td> 461 <p> 462 tgamma(a, z) integer and half integer values 463 </p> 464 </td> 465<td> 466 <p> 467 <span class="blue">Max = 0ε (Mean = 0ε)</span><br> <br> (<span class="emphasis"><em>GSL 468 2.1:</em></span> Max = 118ε (Mean = 12.5ε))<br> (<span class="emphasis"><em>Rmath 469 3.2.3:</em></span> Max = 138ε (Mean = 16.9ε)) 470 </p> 471 </td> 472<td> 473 <p> 474 <span class="blue">Max = 11.1ε (Mean = 2.07ε)</span> 475 </p> 476 </td> 477<td> 478 <p> 479 <span class="blue">Max = 54.7ε (Mean = 6.16ε)</span> 480 </p> 481 </td> 482<td> 483 <p> 484 <span class="blue">Max = 8.72ε (Mean = 1.48ε)</span> 485 </p> 486 </td> 487</tr> 488</tbody> 489</table></div> 490</div> 491<br class="table-break"><div class="table"> 492<a name="math_toolkit.sf_gamma.igamma.table_tgamma_lower"></a><p class="title"><b>Table 8.11. Error rates for tgamma_lower</b></p> 493<div class="table-contents"><table class="table" summary="Error rates for tgamma_lower"> 494<colgroup> 495<col> 496<col> 497<col> 498<col> 499<col> 500</colgroup> 501<thead><tr> 502<th> 503 </th> 504<th> 505 <p> 506 GNU C++ version 7.1.0<br> linux<br> double 507 </p> 508 </th> 509<th> 510 <p> 511 GNU C++ version 7.1.0<br> linux<br> long double 512 </p> 513 </th> 514<th> 515 <p> 516 Sun compiler version 0x5150<br> Sun Solaris<br> long double 517 </p> 518 </th> 519<th> 520 <p> 521 Microsoft Visual C++ version 14.1<br> Win32<br> double 522 </p> 523 </th> 524</tr></thead> 525<tbody> 526<tr> 527<td> 528 <p> 529 tgamma(a, z) medium values 530 </p> 531 </td> 532<td> 533 <p> 534 <span class="blue">Max = 0.833ε (Mean = 0.0315ε)</span><br> 535 <br> (<span class="emphasis"><em>GSL 2.1:</em></span> Max = 0.833ε (Mean = 0.0315ε)) 536 </p> 537 </td> 538<td> 539 <p> 540 <span class="blue">Max = 6.79ε (Mean = 1.46ε)</span> 541 </p> 542 </td> 543<td> 544 <p> 545 <span class="blue">Max = 363ε (Mean = 63.8ε)</span> 546 </p> 547 </td> 548<td> 549 <p> 550 <span class="blue">Max = 5.62ε (Mean = 1.49ε)</span> 551 </p> 552 </td> 553</tr> 554<tr> 555<td> 556 <p> 557 tgamma(a, z) small values 558 </p> 559 </td> 560<td> 561 <p> 562 <span class="blue">Max = 0ε (Mean = 0ε)</span><br> <br> (<span class="emphasis"><em>GSL 563 2.1:</em></span> Max = 0ε (Mean = 0ε)) 564 </p> 565 </td> 566<td> 567 <p> 568 <span class="blue">Max = 1.97ε (Mean = 0.555ε)</span> 569 </p> 570 </td> 571<td> 572 <p> 573 <span class="blue">Max = 1.97ε (Mean = 0.558ε)</span> 574 </p> 575 </td> 576<td> 577 <p> 578 <span class="blue">Max = 1.57ε (Mean = 0.525ε)</span> 579 </p> 580 </td> 581</tr> 582<tr> 583<td> 584 <p> 585 tgamma(a, z) integer and half integer values 586 </p> 587 </td> 588<td> 589 <p> 590 <span class="blue">Max = 0ε (Mean = 0ε)</span><br> <br> (<span class="emphasis"><em>GSL 591 2.1:</em></span> Max = 0ε (Mean = 0ε)) 592 </p> 593 </td> 594<td> 595 <p> 596 <span class="blue">Max = 4.83ε (Mean = 1.15ε)</span> 597 </p> 598 </td> 599<td> 600 <p> 601 <span class="blue">Max = 84.7ε (Mean = 17.5ε)</span> 602 </p> 603 </td> 604<td> 605 <p> 606 <span class="blue">Max = 2.69ε (Mean = 0.849ε)</span> 607 </p> 608 </td> 609</tr> 610</tbody> 611</table></div> 612</div> 613<br class="table-break"><div class="table"> 614<a name="math_toolkit.sf_gamma.igamma.table_tgamma_incomplete_"></a><p class="title"><b>Table 8.12. Error rates for tgamma (incomplete)</b></p> 615<div class="table-contents"><table class="table" summary="Error rates for tgamma (incomplete)"> 616<colgroup> 617<col> 618<col> 619<col> 620<col> 621<col> 622</colgroup> 623<thead><tr> 624<th> 625 </th> 626<th> 627 <p> 628 GNU C++ version 7.1.0<br> linux<br> double 629 </p> 630 </th> 631<th> 632 <p> 633 GNU C++ version 7.1.0<br> linux<br> long double 634 </p> 635 </th> 636<th> 637 <p> 638 Sun compiler version 0x5150<br> Sun Solaris<br> long double 639 </p> 640 </th> 641<th> 642 <p> 643 Microsoft Visual C++ version 14.1<br> Win32<br> double 644 </p> 645 </th> 646</tr></thead> 647<tbody> 648<tr> 649<td> 650 <p> 651 tgamma(a, z) medium values 652 </p> 653 </td> 654<td> 655 <p> 656 <span class="blue">Max = 0ε (Mean = 0ε)</span><br> <br> (<span class="emphasis"><em>GSL 657 2.1:</em></span> Max = 200ε (Mean = 13.3ε)) 658 </p> 659 </td> 660<td> 661 <p> 662 <span class="blue">Max = 8.47ε (Mean = 1.9ε)</span> 663 </p> 664 </td> 665<td> 666 <p> 667 <span class="blue">Max = 412ε (Mean = 95.5ε)</span> 668 </p> 669 </td> 670<td> 671 <p> 672 <span class="blue">Max = 8.14ε (Mean = 1.76ε)</span> 673 </p> 674 </td> 675</tr> 676<tr> 677<td> 678 <p> 679 tgamma(a, z) small values 680 </p> 681 </td> 682<td> 683 <p> 684 <span class="blue">Max = 0.753ε (Mean = 0.0474ε)</span><br> 685 <br> (<span class="emphasis"><em>GSL 2.1:</em></span> <span class="red">Max = 686 1.38e+10ε (Mean = 1.05e+09ε))</span> 687 </p> 688 </td> 689<td> 690 <p> 691 <span class="blue">Max = 2.31ε (Mean = 0.775ε)</span> 692 </p> 693 </td> 694<td> 695 <p> 696 <span class="blue">Max = 2.13ε (Mean = 0.717ε)</span> 697 </p> 698 </td> 699<td> 700 <p> 701 <span class="blue">Max = 2.53ε (Mean = 0.66ε)</span> 702 </p> 703 </td> 704</tr> 705<tr> 706<td> 707 <p> 708 tgamma(a, z) integer and half integer values 709 </p> 710 </td> 711<td> 712 <p> 713 <span class="blue">Max = 0ε (Mean = 0ε)</span><br> <br> (<span class="emphasis"><em>GSL 714 2.1:</em></span> Max = 117ε (Mean = 12.5ε)) 715 </p> 716 </td> 717<td> 718 <p> 719 <span class="blue">Max = 5.52ε (Mean = 1.48ε)</span> 720 </p> 721 </td> 722<td> 723 <p> 724 <span class="blue">Max = 79.6ε (Mean = 20.9ε)</span> 725 </p> 726 </td> 727<td> 728 <p> 729 <span class="blue">Max = 5.16ε (Mean = 1.33ε)</span> 730 </p> 731 </td> 732</tr> 733</tbody> 734</table></div> 735</div> 736<br class="table-break"><h5> 737<a name="math_toolkit.sf_gamma.igamma.h3"></a> 738 <span class="phrase"><a name="math_toolkit.sf_gamma.igamma.testing"></a></span><a class="link" href="igamma.html#math_toolkit.sf_gamma.igamma.testing">Testing</a> 739 </h5> 740<p> 741 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> 742 with this implementation to perform a basic "sanity check". Accuracy 743 tests use data generated at very high precision (using NTL's RR class set 744 at 1000-bit precision) using this implementation with a very high precision 745 60-term <a class="link" href="../lanczos.html" title="The Lanczos Approximation">Lanczos approximation</a>, 746 and some but not all of the special case handling disabled. This is less 747 than satisfactory: an independent method should really be used, but apparently 748 a complete lack of such methods are available. We can't even use a deliberately 749 naive implementation without special case handling since Legendre's continued 750 fraction (see below) is unstable for small a and z. 751 </p> 752<h5> 753<a name="math_toolkit.sf_gamma.igamma.h4"></a> 754 <span class="phrase"><a name="math_toolkit.sf_gamma.igamma.implementation"></a></span><a class="link" href="igamma.html#math_toolkit.sf_gamma.igamma.implementation">Implementation</a> 755 </h5> 756<p> 757 These four functions share a common implementation since they are all related 758 via: 759 </p> 760<p> 761 1) 762 </p> 763<div class="blockquote"><blockquote class="blockquote"><p> 764 <span class="inlinemediaobject"><img src="../../../equations/igamma5.svg"></span> 765 766 </p></blockquote></div> 767<p> 768 2) 769 </p> 770<div class="blockquote"><blockquote class="blockquote"><p> 771 <span class="inlinemediaobject"><img src="../../../equations/igamma6.svg"></span> 772 773 </p></blockquote></div> 774<p> 775 3) 776 </p> 777<div class="blockquote"><blockquote class="blockquote"><p> 778 <span class="inlinemediaobject"><img src="../../../equations/igamma7.svg"></span> 779 780 </p></blockquote></div> 781<p> 782 The lower incomplete gamma is computed from its series representation: 783 </p> 784<p> 785 4) 786 </p> 787<div class="blockquote"><blockquote class="blockquote"><p> 788 <span class="inlinemediaobject"><img src="../../../equations/igamma8.svg"></span> 789 790 </p></blockquote></div> 791<p> 792 Or by subtraction of the upper integral from either Γ(a) or 1 when <span class="emphasis"><em>x 793 - (1</em></span>(3x)) > a and x > 1.1/. 794 </p> 795<p> 796 The upper integral is computed from Legendre's continued fraction representation: 797 </p> 798<p> 799 5) 800 </p> 801<div class="blockquote"><blockquote class="blockquote"><p> 802 <span class="inlinemediaobject"><img src="../../../equations/igamma9.svg"></span> 803 804 </p></blockquote></div> 805<p> 806 When <span class="emphasis"><em>(x > 1.1)</em></span> or by subtraction of the lower integral 807 from either Γ(a) or 1 when <span class="emphasis"><em>x - (1</em></span>(3x)) < a/. 808 </p> 809<p> 810 For <span class="emphasis"><em>x < 1.1</em></span> computation of the upper integral is 811 more complex as the continued fraction representation is unstable in this 812 area. However there is another series representation for the lower integral: 813 </p> 814<p> 815 6) 816 </p> 817<div class="blockquote"><blockquote class="blockquote"><p> 818 <span class="inlinemediaobject"><img src="../../../equations/igamma10.svg"></span> 819 820 </p></blockquote></div> 821<p> 822 That lends itself to calculation of the upper integral via rearrangement 823 to: 824 </p> 825<p> 826 7) 827 </p> 828<div class="blockquote"><blockquote class="blockquote"><p> 829 <span class="inlinemediaobject"><img src="../../../equations/igamma11.svg"></span> 830 831 </p></blockquote></div> 832<p> 833 Refer to the documentation for <a class="link" href="../powers/powm1.html" title="powm1">powm1</a> 834 and <a class="link" href="tgamma.html" title="Gamma">tgamma1pm1</a> for details 835 of their implementation. 836 </p> 837<p> 838 For <span class="emphasis"><em>x < 1.1</em></span> the crossover point where the result 839 is ~0.5 no longer occurs for <span class="emphasis"><em>x ~ y</em></span>. Using <span class="emphasis"><em>x 840 * 0.75 < a</em></span> as the crossover criterion for <span class="emphasis"><em>0.5 < 841 x <= 1.1</em></span> keeps the maximum value computed (whether it's the 842 upper or lower interval) to around 0.75. Likewise for <span class="emphasis"><em>x <= 0.5</em></span> 843 then using <span class="emphasis"><em>-0.4 / log(x) < a</em></span> as the crossover criterion 844 keeps the maximum value computed to around 0.7 (whether it's the upper or 845 lower interval). 846 </p> 847<p> 848 There are two special cases used when a is an integer or half integer, and 849 the crossover conditions listed above indicate that we should compute the 850 upper integral Q. If a is an integer in the range <span class="emphasis"><em>1 <= a < 851 30</em></span> then the following finite sum is used: 852 </p> 853<p> 854 9) 855 </p> 856<div class="blockquote"><blockquote class="blockquote"><p> 857 <span class="inlinemediaobject"><img src="../../../equations/igamma1f.svg"></span> 858 859 </p></blockquote></div> 860<p> 861 While for half-integers in the range <span class="emphasis"><em>0.5 <= a < 30</em></span> 862 then the following finite sum is used: 863 </p> 864<p> 865 10) 866 </p> 867<div class="blockquote"><blockquote class="blockquote"><p> 868 <span class="inlinemediaobject"><img src="../../../equations/igamma2f.svg"></span> 869 870 </p></blockquote></div> 871<p> 872 These are both more stable and more efficient than the continued fraction 873 alternative. 874 </p> 875<p> 876 When the argument <span class="emphasis"><em>a</em></span> is large, and <span class="emphasis"><em>x ~ a</em></span> 877 then the series (4) and continued fraction (5) above are very slow to converge. 878 In this area an expansion due to Temme is used: 879 </p> 880<p> 881 11) 882 </p> 883<div class="blockquote"><blockquote class="blockquote"><p> 884 <span class="inlinemediaobject"><img src="../../../equations/igamma16.svg"></span> 885 886 </p></blockquote></div> 887<p> 888 12) 889 </p> 890<div class="blockquote"><blockquote class="blockquote"><p> 891 <span class="inlinemediaobject"><img src="../../../equations/igamma17.svg"></span> 892 893 </p></blockquote></div> 894<p> 895 13) 896 </p> 897<div class="blockquote"><blockquote class="blockquote"><p> 898 <span class="inlinemediaobject"><img src="../../../equations/igamma18.svg"></span> 899 900 </p></blockquote></div> 901<p> 902 14) 903 </p> 904<div class="blockquote"><blockquote class="blockquote"><p> 905 <span class="inlinemediaobject"><img src="../../../equations/igamma19.svg"></span> 906 907 </p></blockquote></div> 908<p> 909 The double sum is truncated to a fixed number of terms - to give a specific 910 target precision - and evaluated as a polynomial-of-polynomials. There are 911 versions for up to 128-bit long double precision: types requiring greater 912 precision than that do not use these expansions. The coefficients C<sub>k</sub><sup>n</sup> are 913 computed in advance using the recurrence relations given by Temme. The zone 914 where these expansions are used is 915 </p> 916<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> 917</pre> 918<p> 919 And: 920 </p> 921<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> 922</pre> 923<p> 924 The latter range is valid for all types up to 128-bit long doubles, and is 925 designed to ensure that the result is larger than 10<sup>-6</sup>, the first range is 926 used only for types up to 80-bit long doubles. These domains are narrower 927 than the ones recommended by either Temme or Didonato and Morris. However, 928 using a wider range results in large and inexact (i.e. computed) values being 929 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 930 larger error rates. In other words there is a fine trade off here between 931 efficiency and error. The current limits should keep the number of terms 932 required by (4) and (5) to no more than ~20 at double precision. 933 </p> 934<p> 935 For the normalised incomplete gamma functions, calculation of the leading 936 power terms is central to the accuracy of the function. For smallish a and 937 x combining the power terms with the <a class="link" href="../lanczos.html" title="The Lanczos Approximation">Lanczos 938 approximation</a> gives the greatest accuracy: 939 </p> 940<p> 941 15) 942 </p> 943<div class="blockquote"><blockquote class="blockquote"><p> 944 <span class="inlinemediaobject"><img src="../../../equations/igamma12.svg"></span> 945 946 </p></blockquote></div> 947<p> 948 In the event that this causes underflow/overflow then the exponent can be 949 reduced by a factor of <span class="emphasis"><em>a</em></span> and brought inside the power 950 term. 951 </p> 952<p> 953 When a and x are large, we end up with a very large exponent with a base 954 near one: this will not be computed accurately via the pow function, and 955 taking logs simply leads to cancellation errors. The worst of the errors 956 can be avoided by using: 957 </p> 958<p> 959 16) 960 </p> 961<div class="blockquote"><blockquote class="blockquote"><p> 962 <span class="inlinemediaobject"><img src="../../../equations/igamma13.svg"></span> 963 964 </p></blockquote></div> 965<p> 966 when <span class="emphasis"><em>a-x</em></span> is small and a and x are large. There is still 967 a subtraction and therefore some cancellation errors - but the terms are 968 small so the absolute error will be small - and it is absolute rather than 969 relative error that counts in the argument to the <span class="emphasis"><em>exp</em></span> 970 function. Note that for sufficiently large a and x the errors will still 971 get you eventually, although this does delay the inevitable much longer than 972 other methods. Use of <span class="emphasis"><em>log(1+x)-x</em></span> here is inspired by 973 Temme (see references below). 974 </p> 975<h5> 976<a name="math_toolkit.sf_gamma.igamma.h5"></a> 977 <span class="phrase"><a name="math_toolkit.sf_gamma.igamma.references"></a></span><a class="link" href="igamma.html#math_toolkit.sf_gamma.igamma.references">References</a> 978 </h5> 979<div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: disc; "> 980<li class="listitem"> 981 N. M. Temme, A Set of Algorithms for the Incomplete Gamma Functions, 982 Probability in the Engineering and Informational Sciences, 8, 1994. 983 </li> 984<li class="listitem"> 985 N. M. Temme, The Asymptotic Expansion of the Incomplete Gamma Functions, 986 Siam J. Math Anal. Vol 10 No 4, July 1979, p757. 987 </li> 988<li class="listitem"> 989 A. R. Didonato and A. H. Morris, Computation of the Incomplete Gamma 990 Function Ratios and their Inverse. ACM TOMS, Vol 12, No 4, Dec 1986, 991 p377. 992 </li> 993<li class="listitem"> 994 W. Gautschi, The Incomplete Gamma Functions Since Tricomi, In Tricomi's 995 Ideas and Contemporary Applied Mathematics, Atti dei Convegni Lincei, 996 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> 997 </li> 998</ul></div> 999</div> 1000<table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr> 1001<td align="left"></td> 1002<td align="right"><div class="copyright-footer">Copyright © 2006-2019 Nikhar 1003 Agrawal, Anton Bikineev, Paul A. 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(See accompanying 1008 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>) 1009 </p> 1010</div></td> 1011</tr></table> 1012<hr> 1013<div class="spirit-nav"> 1014<a accesskey="p" href="gamma_ratios.html"><img src="../../../../../../doc/src/images/prev.png" alt="Prev"></a><a accesskey="u" href="../sf_gamma.html"><img src="../../../../../../doc/src/images/up.png" alt="Up"></a><a accesskey="h" href="../../index.html"><img src="../../../../../../doc/src/images/home.png" alt="Home"></a><a accesskey="n" href="igamma_inv.html"><img src="../../../../../../doc/src/images/next.png" alt="Next"></a> 1015</div> 1016</body> 1017</html> 1018
