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1 | <html> |
| 2 | <head> | |
| 3 | <meta http-equiv="Content-Type" content="text/html; charset=US-ASCII"> | |
| 4 | <title>Ratios of Gamma Functions</title> | |
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| 11 | </head> | |
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| 13 | <table cellpadding="2" width="100%"><tr> | |
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| 20 | </tr></table> | |
| 21 | <hr> | |
| 22 | <div class="spirit-nav"> | |
| 23 | <a accesskey="p" href="polygamma.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.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.gamma_ratios"></a><a class="link" href="gamma_ratios.html" title="Ratios of Gamma Functions">Ratios of Gamma Functions</a> | |
| 28 | </h3></div></div></div> | |
| 29 | <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> | |
| 30 | </pre> | |
| 31 | <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> | |
| 32 | ||
| 33 | <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> | |
| 34 | <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_ratio</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">b</span><span class="special">);</span> | |
| 35 | ||
| 36 | <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 15. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">></span> | |
| 37 | <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_ratio</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">b</span><span class="special">,</span> <span class="keyword">const</span> <a class="link" href="../../policy.html" title="Chapter 15. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&);</span> | |
| 38 | ||
| 39 | <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> | |
| 40 | <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_delta_ratio</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">delta</span><span class="special">);</span> | |
| 41 | ||
| 42 | <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> | |
| 43 | <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_delta_ratio</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">delta</span><span class="special">,</span> <span class="keyword">const</span> <a class="link" href="../../policy.html" title="Chapter 15. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&);</span> | |
| 44 | ||
| 45 | <span class="special">}}</span> <span class="comment">// namespaces</span> | |
| 46 | </pre> | |
| 47 | <h5> | |
| 48 | <a name="math_toolkit.sf_gamma.gamma_ratios.h0"></a> | |
| 49 | <span class="phrase"><a name="math_toolkit.sf_gamma.gamma_ratios.description"></a></span><a class="link" href="gamma_ratios.html#math_toolkit.sf_gamma.gamma_ratios.description">Description</a> | |
| 50 | </h5> | |
| 51 | <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> | |
| 52 | <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_ratio</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">b</span><span class="special">);</span> | |
| 53 | ||
| 54 | <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 15. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">></span> | |
| 55 | <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_ratio</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">b</span><span class="special">,</span> <span class="keyword">const</span> <a class="link" href="../../policy.html" title="Chapter 15. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&);</span> | |
| 56 | </pre> | |
| 57 | <p> | |
| 58 | Returns the ratio of gamma functions: | |
| 59 | </p> | |
| 60 | <p> | |
| 61 | <span class="inlinemediaobject"><img src="../../../equations/gamma_ratio0.svg"></span> | |
| 62 | </p> | |
| 63 | <p> | |
| 64 | The final <a class="link" href="../../policy.html" title="Chapter 15. Policies: Controlling Precision, Error Handling etc">Policy</a> argument is optional and can | |
| 65 | be used to control the behaviour of the function: how it handles errors, | |
| 66 | what level of precision to use etc. Refer to the <a class="link" href="../../policy.html" title="Chapter 15. Policies: Controlling Precision, Error Handling etc">policy | |
| 67 | documentation for more details</a>. | |
| 68 | </p> | |
| 69 | <p> | |
| 70 | Internally this just calls <code class="computeroutput"><span class="identifier">tgamma_delta_ratio</span><span class="special">(</span><span class="identifier">a</span><span class="special">,</span> | |
| 71 | <span class="identifier">b</span><span class="special">-</span><span class="identifier">a</span><span class="special">)</span></code>. | |
| 72 | </p> | |
| 73 | <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> | |
| 74 | <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_delta_ratio</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">delta</span><span class="special">);</span> | |
| 75 | ||
| 76 | <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 15. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">></span> | |
| 77 | <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_delta_ratio</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">delta</span><span class="special">,</span> <span class="keyword">const</span> <a class="link" href="../../policy.html" title="Chapter 15. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&);</span> | |
| 78 | </pre> | |
| 79 | <p> | |
| 80 | Returns the ratio of gamma functions: | |
| 81 | </p> | |
| 82 | <p> | |
| 83 | <span class="inlinemediaobject"><img src="../../../equations/gamma_ratio1.svg"></span> | |
| 84 | </p> | |
| 85 | <p> | |
| 86 | The final <a class="link" href="../../policy.html" title="Chapter 15. Policies: Controlling Precision, Error Handling etc">Policy</a> argument is optional and can | |
| 87 | be used to control the behaviour of the function: how it handles errors, | |
| 88 | what level of precision to use etc. Refer to the <a class="link" href="../../policy.html" title="Chapter 15. Policies: Controlling Precision, Error Handling etc">policy | |
| 89 | documentation for more details</a>. | |
| 90 | </p> | |
| 91 | <p> | |
| 92 | Note that the result is calculated accurately even when <span class="emphasis"><em>delta</em></span> | |
| 93 | is small compared to <span class="emphasis"><em>a</em></span>: indeed even if <span class="emphasis"><em>a+delta | |
| 94 | ~ a</em></span>. The function is typically used when <span class="emphasis"><em>a</em></span> | |
| 95 | is large and <span class="emphasis"><em>delta</em></span> is very small. | |
| 96 | </p> | |
| 97 | <p> | |
| 98 | 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 | |
| 99 | type calculation rules</em></span></a> when T1 and T2 are different types, | |
| 100 | otherwise the result type is simple T1. | |
| 101 | </p> | |
| 102 | <p> | |
| 103 | <span class="inlinemediaobject"><img src="../../../graphs/tgamma_delta_ratio.svg" align="middle"></span> | |
| 104 | </p> | |
| 105 | <h5> | |
| 106 | <a name="math_toolkit.sf_gamma.gamma_ratios.h1"></a> | |
| 107 | <span class="phrase"><a name="math_toolkit.sf_gamma.gamma_ratios.accuracy"></a></span><a class="link" href="gamma_ratios.html#math_toolkit.sf_gamma.gamma_ratios.accuracy">Accuracy</a> | |
| 108 | </h5> | |
| 109 | <p> | |
| 110 | The following table shows the peak errors (in units of epsilon) found on | |
| 111 | various platforms with various floating point types. Unless otherwise specified | |
| 112 | any floating point type that is narrower than the one shown will have <a class="link" href="../relative_error.html#math_toolkit.relative_error.zero_error">effectively zero error</a>. | |
| 113 | </p> | |
| 114 | <div class="table"> | |
| 115 | <a name="math_toolkit.sf_gamma.gamma_ratios.table_tgamma_delta_ratio"></a><p class="title"><b>Table 6.7. Error rates for tgamma_delta_ratio</b></p> | |
| 116 | <div class="table-contents"><table class="table" summary="Error rates for tgamma_delta_ratio"> | |
| 117 | <colgroup> | |
| 118 | <col> | |
| 119 | <col> | |
| 120 | <col> | |
| 121 | <col> | |
| 122 | <col> | |
| 123 | </colgroup> | |
| 124 | <thead><tr> | |
| 125 | <th> | |
| 126 | </th> | |
| 127 | <th> | |
| 128 | <p> | |
| 129 | Microsoft Visual C++ version 12.0<br> Win32<br> double | |
| 130 | </p> | |
| 131 | </th> | |
| 132 | <th> | |
| 133 | <p> | |
| 134 | GNU C++ version 5.1.0<br> linux<br> double | |
| 135 | </p> | |
| 136 | </th> | |
| 137 | <th> | |
| 138 | <p> | |
| 139 | GNU C++ version 5.1.0<br> linux<br> long double | |
| 140 | </p> | |
| 141 | </th> | |
| 142 | <th> | |
| 143 | <p> | |
| 144 | Sun compiler version 0x5130<br> Sun Solaris<br> long double | |
| 145 | </p> | |
| 146 | </th> | |
| 147 | </tr></thead> | |
| 148 | <tbody> | |
| 149 | <tr> | |
| 150 | <td> | |
| 151 | <p> | |
| 152 | tgamma + small delta ratios | |
| 153 | </p> | |
| 154 | </td> | |
| 155 | <td> | |
| 156 | <p> | |
| 157 | <span class="blue">Max = 10.1ε (Mean = 1.25ε)</span> | |
| 158 | </p> | |
| 159 | </td> | |
| 160 | <td> | |
| 161 | <p> | |
| 162 | <span class="blue">Max = 0ε (Mean = 0ε)</span> | |
| 163 | </p> | |
| 164 | </td> | |
| 165 | <td> | |
| 166 | <p> | |
| 167 | <span class="blue">Max = 5.56ε (Mean = 0.969ε)</span> | |
| 168 | </p> | |
| 169 | </td> | |
| 170 | <td> | |
| 171 | <p> | |
| 172 | <span class="blue">Max = 15.4ε (Mean = 2.09ε)</span> | |
| 173 | </p> | |
| 174 | </td> | |
| 175 | </tr> | |
| 176 | <tr> | |
| 177 | <td> | |
| 178 | <p> | |
| 179 | tgamma + small delta ratios (negative delta) | |
| 180 | </p> | |
| 181 | </td> | |
| 182 | <td> | |
| 183 | <p> | |
| 184 | <span class="blue">Max = 8.04ε (Mean = 1.31ε)</span> | |
| 185 | </p> | |
| 186 | </td> | |
| 187 | <td> | |
| 188 | <p> | |
| 189 | <span class="blue">Max = 0ε (Mean = 0ε)</span> | |
| 190 | </p> | |
| 191 | </td> | |
| 192 | <td> | |
| 193 | <p> | |
| 194 | <span class="blue">Max = 8.67ε (Mean = 1.29ε)</span> | |
| 195 | </p> | |
| 196 | </td> | |
| 197 | <td> | |
| 198 | <p> | |
| 199 | <span class="blue">Max = 18.3ε (Mean = 2.03ε)</span> | |
| 200 | </p> | |
| 201 | </td> | |
| 202 | </tr> | |
| 203 | <tr> | |
| 204 | <td> | |
| 205 | <p> | |
| 206 | tgamma + small integer ratios | |
| 207 | </p> | |
| 208 | </td> | |
| 209 | <td> | |
| 210 | <p> | |
| 211 | <span class="blue">Max = 2.74ε (Mean = 0.736ε)</span> | |
| 212 | </p> | |
| 213 | </td> | |
| 214 | <td> | |
| 215 | <p> | |
| 216 | <span class="blue">Max = 0ε (Mean = 0ε)</span> | |
| 217 | </p> | |
| 218 | </td> | |
| 219 | <td> | |
| 220 | <p> | |
| 221 | <span class="blue">Max = 1.96ε (Mean = 0.677ε)</span> | |
| 222 | </p> | |
| 223 | </td> | |
| 224 | <td> | |
| 225 | <p> | |
| 226 | <span class="blue">Max = 1.96ε (Mean = 0.677ε)</span> | |
| 227 | </p> | |
| 228 | </td> | |
| 229 | </tr> | |
| 230 | <tr> | |
| 231 | <td> | |
| 232 | <p> | |
| 233 | tgamma + small integer ratios (negative delta) | |
| 234 | </p> | |
| 235 | </td> | |
| 236 | <td> | |
| 237 | <p> | |
| 238 | <span class="blue">Max = 2.15ε (Mean = 0.685ε)</span> | |
| 239 | </p> | |
| 240 | </td> | |
| 241 | <td> | |
| 242 | <p> | |
| 243 | <span class="blue">Max = 0ε (Mean = 0ε)</span> | |
| 244 | </p> | |
| 245 | </td> | |
| 246 | <td> | |
| 247 | <p> | |
| 248 | <span class="blue">Max = 1.62ε (Mean = 0.451ε)</span> | |
| 249 | </p> | |
| 250 | </td> | |
| 251 | <td> | |
| 252 | <p> | |
| 253 | <span class="blue">Max = 1.62ε (Mean = 0.451ε)</span> | |
| 254 | </p> | |
| 255 | </td> | |
| 256 | </tr> | |
| 257 | <tr> | |
| 258 | <td> | |
| 259 | <p> | |
| 260 | integer tgamma ratios | |
| 261 | </p> | |
| 262 | </td> | |
| 263 | <td> | |
| 264 | <p> | |
| 265 | <span class="blue">Max = 0.968ε (Mean = 0.386ε)</span> | |
| 266 | </p> | |
| 267 | </td> | |
| 268 | <td> | |
| 269 | <p> | |
| 270 | <span class="blue">Max = 0ε (Mean = 0ε)</span> | |
| 271 | </p> | |
| 272 | </td> | |
| 273 | <td> | |
| 274 | <p> | |
| 275 | <span class="blue">Max = 0.997ε (Mean = 0.4ε)</span> | |
| 276 | </p> | |
| 277 | </td> | |
| 278 | <td> | |
| 279 | <p> | |
| 280 | <span class="blue">Max = 0.997ε (Mean = 0.4ε)</span> | |
| 281 | </p> | |
| 282 | </td> | |
| 283 | </tr> | |
| 284 | <tr> | |
| 285 | <td> | |
| 286 | <p> | |
| 287 | integer tgamma ratios (negative delta) | |
| 288 | </p> | |
| 289 | </td> | |
| 290 | <td> | |
| 291 | <p> | |
| 292 | <span class="blue">Max = 0.974ε (Mean = 0.184ε)</span> | |
| 293 | </p> | |
| 294 | </td> | |
| 295 | <td> | |
| 296 | <p> | |
| 297 | <span class="blue">Max = 0ε (Mean = 0ε)</span> | |
| 298 | </p> | |
| 299 | </td> | |
| 300 | <td> | |
| 301 | <p> | |
| 302 | <span class="blue">Max = 0.853ε (Mean = 0.176ε)</span> | |
| 303 | </p> | |
| 304 | </td> | |
| 305 | <td> | |
| 306 | <p> | |
| 307 | <span class="blue">Max = 0.853ε (Mean = 0.176ε)</span> | |
| 308 | </p> | |
| 309 | </td> | |
| 310 | </tr> | |
| 311 | </tbody> | |
| 312 | </table></div> | |
| 313 | </div> | |
| 314 | <br class="table-break"><div class="table"> | |
| 315 | <a name="math_toolkit.sf_gamma.gamma_ratios.table_tgamma_ratio"></a><p class="title"><b>Table 6.8. Error rates for tgamma_ratio</b></p> | |
| 316 | <div class="table-contents"><table class="table" summary="Error rates for tgamma_ratio"> | |
| 317 | <colgroup> | |
| 318 | <col> | |
| 319 | <col> | |
| 320 | <col> | |
| 321 | <col> | |
| 322 | <col> | |
| 323 | </colgroup> | |
| 324 | <thead><tr> | |
| 325 | <th> | |
| 326 | </th> | |
| 327 | <th> | |
| 328 | <p> | |
| 329 | Microsoft Visual C++ version 12.0<br> Win32<br> double | |
| 330 | </p> | |
| 331 | </th> | |
| 332 | <th> | |
| 333 | <p> | |
| 334 | GNU C++ version 5.1.0<br> linux<br> double | |
| 335 | </p> | |
| 336 | </th> | |
| 337 | <th> | |
| 338 | <p> | |
| 339 | GNU C++ version 5.1.0<br> linux<br> long double | |
| 340 | </p> | |
| 341 | </th> | |
| 342 | <th> | |
| 343 | <p> | |
| 344 | Sun compiler version 0x5130<br> Sun Solaris<br> long double | |
| 345 | </p> | |
| 346 | </th> | |
| 347 | </tr></thead> | |
| 348 | <tbody><tr> | |
| 349 | <td> | |
| 350 | <p> | |
| 351 | tgamma ratios | |
| 352 | </p> | |
| 353 | </td> | |
| 354 | <td> | |
| 355 | <p> | |
| 356 | <span class="blue">Max = 3.66ε (Mean = 1.27ε)</span> | |
| 357 | </p> | |
| 358 | </td> | |
| 359 | <td> | |
| 360 | <p> | |
| 361 | <span class="blue">Max = 0ε (Mean = 0ε)</span> | |
| 362 | </p> | |
| 363 | </td> | |
| 364 | <td> | |
| 365 | <p> | |
| 366 | <span class="blue">Max = 3.09ε (Mean = 1.15ε)</span> | |
| 367 | </p> | |
| 368 | </td> | |
| 369 | <td> | |
| 370 | <p> | |
| 371 | <span class="blue">Max = 174ε (Mean = 61.2ε)</span> | |
| 372 | </p> | |
| 373 | </td> | |
| 374 | </tr></tbody> | |
| 375 | </table></div> | |
| 376 | </div> | |
| 377 | <br class="table-break"><h5> | |
| 378 | <a name="math_toolkit.sf_gamma.gamma_ratios.h2"></a> | |
| 379 | <span class="phrase"><a name="math_toolkit.sf_gamma.gamma_ratios.testing"></a></span><a class="link" href="gamma_ratios.html#math_toolkit.sf_gamma.gamma_ratios.testing">Testing</a> | |
| 380 | </h5> | |
| 381 | <p> | |
| 382 | Accuracy tests use data generated at very high precision (with <a href="http://shoup.net/ntl/doc/RR.txt" target="_top">NTL | |
| 383 | RR class</a> set at 1000-bit precision: about 300 decimal digits) and | |
| 384 | a deliberately naive calculation of Γ(x)/Γ(y). | |
| 385 | </p> | |
| 386 | <h5> | |
| 387 | <a name="math_toolkit.sf_gamma.gamma_ratios.h3"></a> | |
| 388 | <span class="phrase"><a name="math_toolkit.sf_gamma.gamma_ratios.implementation"></a></span><a class="link" href="gamma_ratios.html#math_toolkit.sf_gamma.gamma_ratios.implementation">Implementation</a> | |
| 389 | </h5> | |
| 390 | <p> | |
| 391 | The implementation of these functions is very similar to that of <a class="link" href="../sf_beta/beta_function.html" title="Beta">beta</a>, | |
| 392 | and is based on combining similar power terms to improve accuracy and avoid | |
| 393 | spurious overflow/underflow. | |
| 394 | </p> | |
| 395 | <p> | |
| 396 | In addition there are optimisations for the situation where <span class="emphasis"><em>delta</em></span> | |
| 397 | is a small integer: in which case this function is basically the reciprocal | |
| 398 | of a rising factorial, or where both arguments are smallish integers: in | |
| 399 | which case table lookup of factorials can be used to calculate the ratio. | |
| 400 | </p> | |
| 401 | </div> | |
| 402 | <table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr> | |
| 403 | <td align="left"></td> | |
| 404 | <td align="right"><div class="copyright-footer">Copyright © 2006-2010, 2012-2014 Nikhar Agrawal, | |
| 405 | Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos, Hubert | |
| 406 | Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Johan Råde, Gautam Sewani, | |
| 407 | Benjamin Sobotta, Thijs van den Berg, Daryle Walker and Xiaogang Zhang<p> | |
| 408 | Distributed under the Boost Software License, Version 1.0. (See accompanying | |
| 409 | 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>) | |
| 410 | </p> | |
| 411 | </div></td> | |
| 412 | </tr></table> | |
| 413 | <hr> | |
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