ceph/src/boost/libs/math/doc/html/math_toolkit/sf_gamma/tgamma.html

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  24 </div>
  25 <div class="section">
  26 <div class="titlepage"><div><div><h3 class="title">
  27 <a name="math_toolkit.sf_gamma.tgamma"></a><a class="link" href="tgamma.html" title="Gamma">Gamma</a>
  28 </h3></div></div></div>
  29 <h5>
  30 <a name="math_toolkit.sf_gamma.tgamma.h0"></a>
  31         <span class="phrase"><a name="math_toolkit.sf_gamma.tgamma.synopsis"></a></span><a class="link" href="tgamma.html#math_toolkit.sf_gamma.tgamma.synopsis">Synopsis</a>
  32       </h5>
  33 <pre class="programlisting"><span class="preprocessor">#include</span> <span class="special">&lt;</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">&gt;</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">&lt;</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">&gt;</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">tgamma</span><span class="special">(</span><span class="identifier">T</span> <span class="identifier">z</span><span class="special">);</span>
  39
  40 <span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">,</span> <span class="keyword">class</span> <a class="link" href="../../policy.html" title="Chapter&#160;15.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&gt;</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">tgamma</span><span class="special">(</span><span class="identifier">T</span> <span class="identifier">z</span><span class="special">,</span> <span class="keyword">const</span> <a class="link" href="../../policy.html" title="Chapter&#160;15.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&amp;);</span>
  42
  43 <span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">&gt;</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">tgamma1pm1</span><span class="special">(</span><span class="identifier">T</span> <span class="identifier">dz</span><span class="special">);</span>
  45
  46 <span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">,</span> <span class="keyword">class</span> <a class="link" href="../../policy.html" title="Chapter&#160;15.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&gt;</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">tgamma1pm1</span><span class="special">(</span><span class="identifier">T</span> <span class="identifier">dz</span><span class="special">,</span> <span class="keyword">const</span> <a class="link" href="../../policy.html" title="Chapter&#160;15.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&amp;);</span>
  48
  49 <span class="special">}}</span> <span class="comment">// namespaces</span>
  50 </pre>
  51 <h5>
  52 <a name="math_toolkit.sf_gamma.tgamma.h1"></a>
  53         <span class="phrase"><a name="math_toolkit.sf_gamma.tgamma.description"></a></span><a class="link" href="tgamma.html#math_toolkit.sf_gamma.tgamma.description">Description</a>
  54       </h5>
  55 <pre class="programlisting"><span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">&gt;</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">T</span> <span class="identifier">z</span><span class="special">);</span>
  57
  58 <span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">,</span> <span class="keyword">class</span> <a class="link" href="../../policy.html" title="Chapter&#160;15.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&gt;</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">T</span> <span class="identifier">z</span><span class="special">,</span> <span class="keyword">const</span> <a class="link" href="../../policy.html" title="Chapter&#160;15.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&amp;);</span>
  60 </pre>
  61 <p>
  62         Returns the "true gamma" (hence name tgamma) of value z:
  63       </p>
  64 <p>
  65         <span class="inlinemediaobject"><img src="../../../equations/gamm1.svg"></span>
  66       </p>
  67 <p>
  68         <span class="inlinemediaobject"><img src="../../../graphs/tgamma.svg" align="middle"></span>
  69       </p>
  70 <p>
  71         The final <a class="link" href="../../policy.html" title="Chapter&#160;15.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a> argument is optional and can
  72         be used to control the behaviour of the function: how it handles errors,
  73         what level of precision to use etc. Refer to the <a class="link" href="../../policy.html" title="Chapter&#160;15.&#160;Policies: Controlling Precision, Error Handling etc">policy
  74         documentation for more details</a>.
  75       </p>
  76 <p>
  77         There are effectively two versions of the <a href="http://en.wikipedia.org/wiki/Gamma_function" target="_top">tgamma</a>
  78         function internally: a fully generic version that is slow, but reasonably
  79         accurate, and a much more efficient approximation that is used where the
  80         number of digits in the significand of T correspond to a certain <a class="link" href="../lanczos.html" title="The Lanczos Approximation">Lanczos
  81         approximation</a>. In practice any built in floating point type you will
  82         encounter has an appropriate <a class="link" href="../lanczos.html" title="The Lanczos Approximation">Lanczos
  83         approximation</a> defined for it. It is also possible, given enough machine
  84         time, to generate further <a class="link" href="../lanczos.html" title="The Lanczos Approximation">Lanczos approximation</a>'s
  85         using the program libs/math/tools/lanczos_generator.cpp.
  86       </p>
  87 <p>
  88         The return type of this function is computed using the <a class="link" href="../result_type.html" title="Calculation of the Type of the Result"><span class="emphasis"><em>result
  89         type calculation rules</em></span></a>: the result is <code class="computeroutput"><span class="keyword">double</span></code>
  90         when T is an integer type, and T otherwise.
  91       </p>
  92 <pre class="programlisting"><span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">&gt;</span>
  93 <a class="link" href="../result_type.html" title="Calculation of the Type of the Result"><span class="emphasis"><em>calculated-result-type</em></span></a> <span class="identifier">tgamma1pm1</span><span class="special">(</span><span class="identifier">T</span> <span class="identifier">dz</span><span class="special">);</span>
  94
  95 <span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">,</span> <span class="keyword">class</span> <a class="link" href="../../policy.html" title="Chapter&#160;15.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&gt;</span>
  96 <a class="link" href="../result_type.html" title="Calculation of the Type of the Result"><span class="emphasis"><em>calculated-result-type</em></span></a> <span class="identifier">tgamma1pm1</span><span class="special">(</span><span class="identifier">T</span> <span class="identifier">dz</span><span class="special">,</span> <span class="keyword">const</span> <a class="link" href="../../policy.html" title="Chapter&#160;15.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&amp;);</span>
  97 </pre>
  98 <p>
  99         Returns <code class="computeroutput"><span class="identifier">tgamma</span><span class="special">(</span><span class="identifier">dz</span> <span class="special">+</span> <span class="number">1</span><span class="special">)</span> <span class="special">-</span> <span class="number">1</span></code>.
 100         Internally the implementation does not make use of the addition and subtraction
 101         implied by the definition, leading to accurate results even for very small
 102         <code class="computeroutput"><span class="identifier">dz</span></code>. However, the implementation
 103         is capped to either 35 digit accuracy, or to the precision of the <a class="link" href="../lanczos.html" title="The Lanczos Approximation">Lanczos
 104         approximation</a> associated with type T, whichever is more accurate.
 105       </p>
 106 <p>
 107         The return type of this function is computed using the <a class="link" href="../result_type.html" title="Calculation of the Type of the Result"><span class="emphasis"><em>result
 108         type calculation rules</em></span></a>: the result is <code class="computeroutput"><span class="keyword">double</span></code>
 109         when T is an integer type, and T otherwise.
 110       </p>
 111 <p>
 112         The final <a class="link" href="../../policy.html" title="Chapter&#160;15.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a> argument is optional and can
 113         be used to control the behaviour of the function: how it handles errors,
 114         what level of precision to use etc. Refer to the <a class="link" href="../../policy.html" title="Chapter&#160;15.&#160;Policies: Controlling Precision, Error Handling etc">policy
 115         documentation for more details</a>.
 116       </p>
 117 <h5>
 118 <a name="math_toolkit.sf_gamma.tgamma.h2"></a>
 119         <span class="phrase"><a name="math_toolkit.sf_gamma.tgamma.accuracy"></a></span><a class="link" href="tgamma.html#math_toolkit.sf_gamma.tgamma.accuracy">Accuracy</a>
 120       </h5>
 121 <p>
 122         The following table shows the peak errors (in units of epsilon) found on
 123         various platforms with various floating point types, along with comparisons
 124         to other common libraries. Unless otherwise specified any floating point
 125         type that is narrower than the one shown will have <a class="link" href="../relative_error.html#math_toolkit.relative_error.zero_error">effectively
 126         zero error</a>.
 127       </p>
 128 <div class="table">
 129 <a name="math_toolkit.sf_gamma.tgamma.table_tgamma"></a><p class="title"><b>Table&#160;6.1.&#160;Error rates for tgamma</b></p>
 130 <div class="table-contents"><table class="table" summary="Error rates for tgamma">
 131 <colgroup>
 132 <col>
 133 <col>
 134 <col>
 135 <col>
 136 <col>
 137 </colgroup>
 138 <thead><tr>
 139 <th>
 140               </th>
 141 <th>
 142                 <p>
 143                   Microsoft Visual C++ version 12.0<br> Win32<br> double
 144                 </p>
 145               </th>
 146 <th>
 147                 <p>
 148                   GNU C++ version 5.1.0<br> linux<br> double
 149                 </p>
 150               </th>
 151 <th>
 152                 <p>
 153                   GNU C++ version 5.1.0<br> linux<br> long double
 154                 </p>
 155               </th>
 156 <th>
 157                 <p>
 158                   Sun compiler version 0x5130<br> Sun Solaris<br> long double
 159                 </p>
 160               </th>
 161 </tr></thead>
 162 <tbody>
 163 <tr>
 164 <td>
 165                 <p>
 166                   factorials
 167                 </p>
 168               </td>
 169 <td>
 170                 <p>
 171                   <span class="blue">Max = 1.85&#949; (Mean = 0.491&#949;)</span><br> <br>
 172                   (<span class="emphasis"><em>&lt;math.h&gt;:</em></span> Max = 3.17&#949; (Mean = 0.928&#949;))
 173                 </p>
 174               </td>
 175 <td>
 176                 <p>
 177                   <span class="blue">Max = 0&#949; (Mean = 0&#949;)</span><br> <br> (<span class="emphasis"><em>GSL
 178                   1.16:</em></span> Max = 3.95&#949; (Mean = 0.783&#949;))<br> (<span class="emphasis"><em>Rmath
 179                   3.0.2:</em></span> Max = 314&#949; (Mean = 93.4&#949;))<br> (<span class="emphasis"><em>Cephes:</em></span>
 180                   Max = 3.19&#949; (Mean = 0.884&#949;))
 181                 </p>
 182               </td>
 183 <td>
 184                 <p>
 185                   <span class="blue">Max = 1.96&#949; (Mean = 0.483&#949;)</span><br> <br>
 186                   (<span class="emphasis"><em>&lt;tr1/cmath&gt;:</em></span> Max = 1.66&#949; (Mean = 0.584&#949;))<br>
 187                   (<span class="emphasis"><em>&lt;math.h&gt;:</em></span> Max = 1.66&#949; (Mean = 0.584&#949;))
 188                 </p>
 189               </td>
 190 <td>
 191                 <p>
 192                   <span class="blue">Max = 172&#949; (Mean = 41&#949;)</span><br> <br>
 193                   (<span class="emphasis"><em>&lt;math.h&gt;:</em></span> Max = 0&#949; (Mean = 0&#949;))
 194                 </p>
 195               </td>
 196 </tr>
 197 <tr>
 198 <td>
 199                 <p>
 200                   near 0
 201                 </p>
 202               </td>
 203 <td>
 204                 <p>
 205                   <span class="blue">Max = 1.96&#949; (Mean = 0.684&#949;)</span><br> <br>
 206                   (<span class="emphasis"><em>&lt;math.h&gt;:</em></span> Max = 1&#949; (Mean = 0.405&#949;))
 207                 </p>
 208               </td>
 209 <td>
 210                 <p>
 211                   <span class="blue">Max = 0&#949; (Mean = 0&#949;)</span><br> <br> (<span class="emphasis"><em>GSL
 212                   1.16:</em></span> Max = 4.51&#949; (Mean = 1.92&#949;))<br> (<span class="emphasis"><em>Rmath
 213                   3.0.2:</em></span> Max = 1&#949; (Mean = 0.335&#949;))<br> (<span class="emphasis"><em>Cephes:</em></span>
 214                   Max = 1&#949; (Mean = 0.548&#949;))
 215                 </p>
 216               </td>
 217 <td>
 218                 <p>
 219                   <span class="blue">Max = 2&#949; (Mean = 0.73&#949;)</span><br> <br>
 220                   (<span class="emphasis"><em>&lt;tr1/cmath&gt;:</em></span> Max = 1&#949; (Mean = 0.376&#949;))<br>
 221                   (<span class="emphasis"><em>&lt;math.h&gt;:</em></span> Max = 1&#949; (Mean = 0.376&#949;))
 222                 </p>
 223               </td>
 224 <td>
 225                 <p>
 226                   <span class="blue">Max = 2&#949; (Mean = 0.647&#949;)</span><br> <br>
 227                   (<span class="emphasis"><em>&lt;math.h&gt;:</em></span> Max = 0.5&#949; (Mean = 0.0791&#949;))
 228                 </p>
 229               </td>
 230 </tr>
 231 <tr>
 232 <td>
 233                 <p>
 234                   near 1
 235                 </p>
 236               </td>
 237 <td>
 238                 <p>
 239                   <span class="blue">Max = 2&#949; (Mean = 0.865&#949;)</span><br> <br>
 240                   (<span class="emphasis"><em>&lt;math.h&gt;:</em></span> Max = 1&#949; (Mean = 0.4&#949;))
 241                 </p>
 242               </td>
 243 <td>
 244                 <p>
 245                   <span class="blue">Max = 0&#949; (Mean = 0&#949;)</span><br> <br> (<span class="emphasis"><em>GSL
 246                   1.16:</em></span> Max = 4.41&#949; (Mean = 1.81&#949;))<br> (<span class="emphasis"><em>Rmath
 247                   3.0.2:</em></span> Max = 1&#949; (Mean = 0.32&#949;))<br> (<span class="emphasis"><em>Cephes:</em></span>
 248                   Max = 1&#949; (Mean = 0.518&#949;))
 249                 </p>
 250               </td>
 251 <td>
 252                 <p>
 253                   <span class="blue">Max = 2&#949; (Mean = 0.85&#949;)</span><br> <br>
 254                   (<span class="emphasis"><em>&lt;tr1/cmath&gt;:</em></span> Max = 0.918&#949; (Mean = 0.203&#949;))<br>
 255                   (<span class="emphasis"><em>&lt;math.h&gt;:</em></span> Max = 0.918&#949; (Mean = 0.203&#949;))
 256                 </p>
 257               </td>
 258 <td>
 259                 <p>
 260                   <span class="blue">Max = 3.01&#949; (Mean = 1.06&#949;)</span><br> <br>
 261                   (<span class="emphasis"><em>&lt;math.h&gt;:</em></span> Max = 1&#949; (Mean = 0.175&#949;))
 262                 </p>
 263               </td>
 264 </tr>
 265 <tr>
 266 <td>
 267                 <p>
 268                   near 2
 269                 </p>
 270               </td>
 271 <td>
 272                 <p>
 273                   <span class="blue">Max = 2&#949; (Mean = 0.995&#949;)</span><br> <br>
 274                   (<span class="emphasis"><em>&lt;math.h&gt;:</em></span> Max = 0&#949; (Mean = 0&#949;))
 275                 </p>
 276               </td>
 277 <td>
 278                 <p>
 279                   <span class="blue">Max = 0&#949; (Mean = 0&#949;)</span><br> <br> (<span class="emphasis"><em>GSL
 280                   1.16:</em></span> Max = 7.95&#949; (Mean = 3.12&#949;))<br> (<span class="emphasis"><em>Rmath
 281                   3.0.2:</em></span> Max = 1&#949; (Mean = 0.191&#949;))<br> (<span class="emphasis"><em>Cephes:</em></span>
 282                   Max = 1.09&#949; (Mean = 0.502&#949;))
 283                 </p>
 284               </td>
 285 <td>
 286                 <p>
 287                   <span class="blue">Max = 2&#949; (Mean = 0.913&#949;)</span><br> <br>
 288                   (<span class="emphasis"><em>&lt;tr1/cmath&gt;:</em></span> Max = 0.558&#949; (Mean = 0.298&#949;))<br>
 289                   (<span class="emphasis"><em>&lt;math.h&gt;:</em></span> Max = 0.558&#949; (Mean = 0.298&#949;))
 290                 </p>
 291               </td>
 292 <td>
 293                 <p>
 294                   <span class="blue">Max = 5.01&#949; (Mean = 1.89&#949;)</span><br> <br>
 295                   (<span class="emphasis"><em>&lt;math.h&gt;:</em></span> Max = 0&#949; (Mean = 0&#949;))
 296                 </p>
 297               </td>
 298 </tr>
 299 <tr>
 300 <td>
 301                 <p>
 302                   near -10
 303                 </p>
 304               </td>
 305 <td>
 306                 <p>
 307                   <span class="blue">Max = 1.73&#949; (Mean = 0.729&#949;)</span><br> <br>
 308                   (<span class="emphasis"><em>&lt;math.h&gt;:</em></span> Max = 0.866&#949; (Mean = 0.445&#949;))
 309                 </p>
 310               </td>
 311 <td>
 312                 <p>
 313                   <span class="blue">Max = 0&#949; (Mean = 0&#949;)</span><br> <br> (<span class="emphasis"><em>GSL
 314                   1.16:</em></span> Max = 2.6&#949; (Mean = 1.05&#949;))<br> (<span class="emphasis"><em>Rmath
 315                   3.0.2:</em></span> Max = 6.34e+05&#949; (Mean = 1.2e+05&#949;))<br> (<span class="emphasis"><em>Cephes:</em></span>
 316                   Max = 2.6&#949; (Mean = 0.956&#949;))
 317                 </p>
 318               </td>
 319 <td>
 320                 <p>
 321                   <span class="blue">Max = 2.6&#949; (Mean = 0.985&#949;)</span><br> <br>
 322                   (<span class="emphasis"><em>&lt;tr1/cmath&gt;:</em></span> Max = 2.26&#949; (Mean = 1.08&#949;))<br>
 323                   (<span class="emphasis"><em>&lt;math.h&gt;:</em></span> Max = 2.26&#949; (Mean = 1.08&#949;))
 324                 </p>
 325               </td>
 326 <td>
 327                 <p>
 328                   <span class="blue">Max = 1.75&#949; (Mean = 0.819&#949;)</span><br> <br>
 329                   (<span class="emphasis"><em>&lt;math.h&gt;:</em></span> Max = 0&#949; (Mean = 0&#949;))
 330                 </p>
 331               </td>
 332 </tr>
 333 <tr>
 334 <td>
 335                 <p>
 336                   near -55
 337                 </p>
 338               </td>
 339 <td>
 340                 <p>
 341                   <span class="blue">Max = 1.8&#949; (Mean = 0.817&#949;)</span><br> <br>
 342                   (<span class="emphasis"><em>&lt;math.h&gt;:</em></span> Max = 3.87e+004&#949; (Mean = 6.71e+003&#949;))
 343                 </p>
 344               </td>
 345 <td>
 346                 <p>
 347                   <span class="blue">Max = 0&#949; (Mean = 0&#949;)</span><br> <br> (<span class="emphasis"><em>GSL
 348                   1.16:</em></span> Max = 1.8&#949; (Mean = 0.782&#949;))<br> (<span class="emphasis"><em>Rmath
 349                   3.0.2:</em></span> Max = 6.36e+06&#949; (Mean = 1.13e+06&#949;))<br> (<span class="emphasis"><em>Cephes:</em></span>
 350                   Max = 2.7&#949; (Mean = 0.988&#949;))
 351                 </p>
 352               </td>
 353 <td>
 354                 <p>
 355                   <span class="blue">Max = 1.8&#949; (Mean = 0.847&#949;)</span><br> <br>
 356                   (<span class="emphasis"><em>&lt;tr1/cmath&gt;:</em></span> Max = 1.79&#949; (Mean = 0.75&#949;))<br>
 357                   (<span class="emphasis"><em>&lt;math.h&gt;:</em></span> Max = 1.79&#949; (Mean = 0.75&#949;))
 358                 </p>
 359               </td>
 360 <td>
 361                 <p>
 362                   <span class="blue">Max = 98.5&#949; (Mean = 53.4&#949;)</span><br> <br>
 363                   (<span class="emphasis"><em>&lt;math.h&gt;:</em></span> Max = 0&#949; (Mean = 0&#949;))
 364                 </p>
 365               </td>
 366 </tr>
 367 </tbody>
 368 </table></div>
 369 </div>
 370 <br class="table-break"><div class="table">
 371 <a name="math_toolkit.sf_gamma.tgamma.table_tgamma1pm1"></a><p class="title"><b>Table&#160;6.2.&#160;Error rates for tgamma1pm1</b></p>
 372 <div class="table-contents"><table class="table" summary="Error rates for tgamma1pm1">
 373 <colgroup>
 374 <col>
 375 <col>
 376 <col>
 377 <col>
 378 <col>
 379 </colgroup>
 380 <thead><tr>
 381 <th>
 382               </th>
 383 <th>
 384                 <p>
 385                   Microsoft Visual C++ version 12.0<br> Win32<br> double
 386                 </p>
 387               </th>
 388 <th>
 389                 <p>
 390                   GNU C++ version 5.1.0<br> linux<br> double
 391                 </p>
 392               </th>
 393 <th>
 394                 <p>
 395                   GNU C++ version 5.1.0<br> linux<br> long double
 396                 </p>
 397               </th>
 398 <th>
 399                 <p>
 400                   Sun compiler version 0x5130<br> Sun Solaris<br> long double
 401                 </p>
 402               </th>
 403 </tr></thead>
 404 <tbody><tr>
 405 <td>
 406                 <p>
 407                   tgamma1pm1(dz)
 408                 </p>
 409               </td>
 410 <td>
 411                 <p>
 412                   <span class="blue">Max = 0.982&#949; (Mean = 0.399&#949;)</span>
 413                 </p>
 414               </td>
 415 <td>
 416                 <p>
 417                   <span class="blue">Max = 0&#949; (Mean = 0&#949;)</span>
 418                 </p>
 419               </td>
 420 <td>
 421                 <p>
 422                   <span class="blue">Max = 1.12&#949; (Mean = 0.49&#949;)</span>
 423                 </p>
 424               </td>
 425 <td>
 426                 <p>
 427                   <span class="blue">Max = 3.97&#949; (Mean = 0.713&#949;)</span>
 428                 </p>
 429               </td>
 430 </tr></tbody>
 431 </table></div>
 432 </div>
 433 <br class="table-break"><h5>
 434 <a name="math_toolkit.sf_gamma.tgamma.h3"></a>
 435         <span class="phrase"><a name="math_toolkit.sf_gamma.tgamma.testing"></a></span><a class="link" href="tgamma.html#math_toolkit.sf_gamma.tgamma.testing">Testing</a>
 436       </h5>
 437 <p>
 438         The gamma is relatively easy to test: factorials and half-integer factorials
 439         can be calculated exactly by other means and compared with the gamma function.
 440         In addition, some accuracy tests in known tricky areas were computed at high
 441         precision using the generic version of this function.
 442       </p>
 443 <p>
 444         The function <code class="computeroutput"><span class="identifier">tgamma1pm1</span></code> is
 445         tested against values calculated very naively using the formula <code class="computeroutput"><span class="identifier">tgamma</span><span class="special">(</span><span class="number">1</span><span class="special">+</span><span class="identifier">dz</span><span class="special">)-</span><span class="number">1</span></code> with a
 446         lanczos approximation accurate to around 100 decimal digits.
 447       </p>
 448 <h5>
 449 <a name="math_toolkit.sf_gamma.tgamma.h4"></a>
 450         <span class="phrase"><a name="math_toolkit.sf_gamma.tgamma.implementation"></a></span><a class="link" href="tgamma.html#math_toolkit.sf_gamma.tgamma.implementation">Implementation</a>
 451       </h5>
 452 <p>
 453         The generic version of the <code class="computeroutput"><span class="identifier">tgamma</span></code>
 454         function is implemented Sterling's approximation for lgamma for large z:
 455       </p>
 456 <p>
 457         <span class="inlinemediaobject"><img src="../../../equations/gamma6.svg"></span>
 458       </p>
 459 <p>
 460         Following exponentiation, downward recursion is then used for small values
 461         of z.
 462       </p>
 463 <p>
 464         For types of known precision the <a class="link" href="../lanczos.html" title="The Lanczos Approximation">Lanczos
 465         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>
 466         maps type T to an appropriate approximation.
 467       </p>
 468 <p>
 469         For z in the range -20 &lt; z &lt; 1 then recursion is used to shift to z
 470         &gt; 1 via:
 471       </p>
 472 <p>
 473         <span class="inlinemediaobject"><img src="../../../equations/gamm3.svg"></span>
 474       </p>
 475 <p>
 476         For very small z, this helps to preserve the identity:
 477       </p>
 478 <p>
 479         <span class="inlinemediaobject"><img src="../../../equations/gamm4.svg"></span>
 480       </p>
 481 <p>
 482         For z &lt; -20 the reflection formula:
 483       </p>
 484 <p>
 485         <span class="inlinemediaobject"><img src="../../../equations/gamm5.svg"></span>
 486       </p>
 487 <p>
 488         is used. Particular care has to be taken to evaluate the <code class="literal">z * sin(&#960; &#160; *
 489         z)</code> part: a special routine is used to reduce z prior to multiplying
 490         by &#960; &#160; to ensure that the result in is the range [0, &#960;/2]. Without this an excessive
 491         amount of error occurs in this region (which is hard enough already, as the
 492         rate of change near a negative pole is <span class="emphasis"><em>exceptionally</em></span>
 493         high).
 494       </p>
 495 <p>
 496         Finally if the argument is a small integer then table lookup of the factorial
 497         is used.
 498       </p>
 499 <p>
 500         The function <code class="computeroutput"><span class="identifier">tgamma1pm1</span></code> is
 501         implemented using rational approximations <a class="link" href="../sf_implementation.html#math_toolkit.sf_implementation.rational_approximations_used">devised
 502         by JM</a> in the region <code class="computeroutput"><span class="special">-</span><span class="number">0.5</span> <span class="special">&lt;</span> <span class="identifier">dz</span>
 503         <span class="special">&lt;</span> <span class="number">2</span></code>.
 504         These are the same approximations (and internal routines) that are used for
 505         <a class="link" href="lgamma.html" title="Log Gamma">lgamma</a>, and so aren't
 506         detailed further here. The result of the approximation is <code class="computeroutput"><span class="identifier">log</span><span class="special">(</span><span class="identifier">tgamma</span><span class="special">(</span><span class="identifier">dz</span><span class="special">+</span><span class="number">1</span><span class="special">))</span></code> which can
 507         fed into <a class="link" href="../powers/expm1.html" title="expm1">expm1</a> to give the
 508         desired result. Outside the range <code class="computeroutput"><span class="special">-</span><span class="number">0.5</span> <span class="special">&lt;</span> <span class="identifier">dz</span>
 509         <span class="special">&lt;</span> <span class="number">2</span></code>
 510         then the naive formula <code class="computeroutput"><span class="identifier">tgamma1pm1</span><span class="special">(</span><span class="identifier">dz</span><span class="special">)</span>
 511         <span class="special">=</span> <span class="identifier">tgamma</span><span class="special">(</span><span class="identifier">dz</span><span class="special">+</span><span class="number">1</span><span class="special">)-</span><span class="number">1</span></code>
 512         can be used directly.
 513       </p>
 514 </div>
 515 <table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr>
 516 <td align="left"></td>
 517 <td align="right"><div class="copyright-footer">Copyright &#169; 2006-2010, 2012-2014 Nikhar Agrawal,
 518       Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos, Hubert
 519       Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Johan R&#229;de, Gautam Sewani,
 520       Benjamin Sobotta, Thijs van den Berg, Daryle Walker and Xiaogang Zhang<p>
 521         Distributed under the Boost Software License, Version 1.0. (See accompanying
 522         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>)
 523       </p>
 524 </div></td>
 525 </tr></table>
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