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  25 <div class="section">
  26 <div class="titlepage"><div><div><h4 class="title">
  27 <a name="math_toolkit.dist_ref.dists.inverse_chi_squared_dist"></a><a class="link" href="inverse_chi_squared_dist.html" title="Inverse Chi Squared Distribution">Inverse
  28         Chi Squared Distribution</a>
  29 </h4></div></div></div>
  30 <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">distributions</span><span class="special">/</span><span class="identifier">inverse_chi_squared</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">&gt;</span></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">&lt;</span><span class="keyword">class</span> <span class="identifier">RealType</span> <span class="special">=</span> <span class="keyword">double</span><span class="special">,</span>
  34           <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">=</span> <a class="link" href="../../pol_ref/pol_ref_ref.html" title="Policy Class Reference">policies::policy&lt;&gt;</a> <span class="special">&gt;</span>
  35 <span class="keyword">class</span> <span class="identifier">inverse_chi_squared_distribution</span>
  36 <span class="special">{</span>
  37 <span class="keyword">public</span><span class="special">:</span>
  38    <span class="keyword">typedef</span> <span class="identifier">RealType</span> <span class="identifier">value_type</span><span class="special">;</span>
  39    <span class="keyword">typedef</span> <span class="identifier">Policy</span>   <span class="identifier">policy_type</span><span class="special">;</span>
  40
  41    <span class="identifier">inverse_chi_squared_distribution</span><span class="special">(</span><span class="identifier">RealType</span> <span class="identifier">df</span> <span class="special">=</span> <span class="number">1</span><span class="special">);</span> <span class="comment">// Not explicitly scaled, default 1/df.</span>
  42    <span class="identifier">inverse_chi_squared_distribution</span><span class="special">(</span><span class="identifier">RealType</span> <span class="identifier">df</span><span class="special">,</span> <span class="identifier">RealType</span> <span class="identifier">scale</span> <span class="special">=</span> <span class="number">1</span><span class="special">/</span><span class="identifier">df</span><span class="special">);</span>  <span class="comment">// Scaled.</span>
  43
  44    <span class="identifier">RealType</span> <span class="identifier">degrees_of_freedom</span><span class="special">()</span><span class="keyword">const</span><span class="special">;</span> <span class="comment">// Default 1.</span>
  45    <span class="identifier">RealType</span> <span class="identifier">scale</span><span class="special">()</span><span class="keyword">const</span><span class="special">;</span> <span class="comment">// Optional scale [xi] (variance), default 1/degrees_of_freedom.</span>
  46 <span class="special">};</span>
  47
  48 <span class="special">}}</span> <span class="comment">// namespace boost // namespace math</span>
  49 </pre>
  50 <p>
  51           The inverse chi squared distribution is a continuous probability distribution
  52           of the <span class="bold"><strong>reciprocal</strong></span> of a variable distributed
  53           according to the chi squared distribution.
  54         </p>
  55 <p>
  56           The sources below give confusingly different formulae using different symbols
  57           for the distribution pdf, but they are all the same, or related by a change
  58           of variable, or choice of scale.
  59         </p>
  60 <p>
  61           Two constructors are available to implement both the scaled and (implicitly)
  62           unscaled versions.
  63         </p>
  64 <p>
  65           The main version has an explicit scale parameter which implements the
  66           <a href="http://en.wikipedia.org/wiki/Scaled-inverse-chi-square_distribution" target="_top">scaled
  67           inverse chi_squared distribution</a>.
  68         </p>
  69 <p>
  70           A second version has an implicit scale = 1/degrees of freedom and gives
  71           the 1st definition in the <a href="http://en.wikipedia.org/wiki/Inverse-chi-square_distribution" target="_top">Wikipedia
  72           inverse chi_squared distribution</a>. The 2nd Wikipedia inverse chi_squared
  73           distribution definition can be implemented by explicitly specifying a scale
  74           = 1.
  75         </p>
  76 <p>
  77           Both definitions are also available in Wolfram Mathematica and in <a href="http://www.r-project.org/" target="_top">The R Project for Statistical Computing</a>
  78           (geoR) with default scale = 1/degrees of freedom.
  79         </p>
  80 <p>
  81           See
  82         </p>
  83 <div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: disc; ">
  84 <li class="listitem">
  85               Inverse chi_squared distribution <a href="http://en.wikipedia.org/wiki/Inverse-chi-square_distribution" target="_top">http://en.wikipedia.org/wiki/Inverse-chi-square_distribution</a>
  86             </li>
  87 <li class="listitem">
  88               Scaled inverse chi_squared distribution<a href="http://en.wikipedia.org/wiki/Scaled-inverse-chi-square_distribution" target="_top">http://en.wikipedia.org/wiki/Scaled-inverse-chi-square_distribution</a>
  89             </li>
  90 <li class="listitem">
  91               R inverse chi_squared distribution functions <a href="http://hosho.ees.hokudai.ac.jp/~kubo/Rdoc/library/geoR/html/InvChisquare.html" target="_top">R
  92               </a>
  93             </li>
  94 <li class="listitem">
  95               Inverse chi_squared distribution functions <a href="http://mathworld.wolfram.com/InverseChi-SquaredDistribution.html" target="_top">Weisstein,
  96               Eric W. "Inverse Chi-Squared Distribution." From MathWorld--A
  97               Wolfram Web Resource.</a>
  98             </li>
  99 <li class="listitem">
 100               Inverse chi_squared distribution reference <a href="http://reference.wolfram.com/mathematica/ref/InverseChiSquareDistribution.html" target="_top">Weisstein,
 101               Eric W. "Inverse Chi-Squared Distribution reference." From
 102               Wolfram Mathematica.</a>
 103             </li>
 104 </ul></div>
 105 <p>
 106           The inverse_chi_squared distribution is used in <a href="http://en.wikipedia.org/wiki/Bayesian_statistics" target="_top">Bayesian
 107           statistics</a>: the scaled inverse chi-square is conjugate prior for
 108           the normal distribution with known mean, model parameter &#963;&#178; (variance).
 109         </p>
 110 <p>
 111           See <a href="http://en.wikipedia.org/wiki/Conjugate_prior" target="_top">conjugate
 112           priors including a table of distributions and their priors.</a>
 113         </p>
 114 <p>
 115           See also <a class="link" href="inverse_gamma_dist.html" title="Inverse Gamma Distribution">Inverse
 116           Gamma Distribution</a> and <a class="link" href="chi_squared_dist.html" title="Chi Squared Distribution">Chi
 117           Squared Distribution</a>.
 118         </p>
 119 <p>
 120           The inverse_chi_squared distribution is a special case of a inverse_gamma
 121           distribution with &#957; (degrees_of_freedom) shape (&#945;) and scale (&#946;) where
 122         </p>
 123 <p>
 124           &#8192;&#8192; &#945;= &#957; /2 and &#946; = &#189;.
 125         </p>
 126 <div class="note"><table border="0" summary="Note">
 127 <tr>
 128 <td rowspan="2" align="center" valign="top" width="25"><img alt="[Note]" src="../../../../../../../doc/src/images/note.png"></td>
 129 <th align="left">Note</th>
 130 </tr>
 131 <tr><td align="left" valign="top">
 132 <p>
 133             This distribution <span class="bold"><strong>does</strong></span> provide the typedef:
 134           </p>
 135 <pre class="programlisting"><span class="keyword">typedef</span> <span class="identifier">inverse_chi_squared_distribution</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;</span> <span class="identifier">inverse_chi_squared</span><span class="special">;</span></pre>
 136 <p>
 137             If you want a <code class="computeroutput"><span class="keyword">double</span></code> precision
 138             inverse_chi_squared distribution you can use
 139           </p>
 140 <pre class="programlisting"><span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">::</span><span class="identifier">inverse_chi_squared_distribution</span><span class="special">&lt;&gt;</span></pre>
 141 <p>
 142             or you can write <code class="computeroutput"><span class="identifier">inverse_chi_squared</span>
 143             <span class="identifier">my_invchisqr</span><span class="special">(</span><span class="number">2</span><span class="special">,</span> <span class="number">3</span><span class="special">);</span></code>
 144           </p>
 145 </td></tr>
 146 </table></div>
 147 <p>
 148           For degrees of freedom parameter &#957;, the (<span class="bold"><strong>unscaled</strong></span>)
 149           inverse chi_squared distribution is defined by the probability density
 150           function (PDF):
 151         </p>
 152 <p>
 153           &#8192;&#8192; f(x;&#957;) = 2<sup>-&#957;/2</sup> x<sup>-&#957;/2-1</sup> e<sup>-1/2x</sup> / &#915;(&#957;/2)
 154         </p>
 155 <p>
 156           and Cumulative Density Function (CDF)
 157         </p>
 158 <p>
 159           &#8192;&#8192;  F(x;&#957;) = &#915;(&#957;/2, 1/2x) / &#915;(&#957;/2)
 160         </p>
 161 <p>
 162           For degrees of freedom parameter &#957; and scale parameter &#958;, the <span class="bold"><strong>scaled</strong></span>
 163           inverse chi_squared distribution is defined by the probability density
 164           function (PDF):
 165         </p>
 166 <p>
 167           &#8192;&#8192; f(x;&#957;, &#958;) = (&#958;&#957;/2)<sup>&#957;/2</sup> e<sup>-&#957;&#958;/2x</sup> x<sup>-1-&#957;/2</sup> / &#915;(&#957;/2)
 168         </p>
 169 <p>
 170           and Cumulative Density Function (CDF)
 171         </p>
 172 <p>
 173           &#8192;&#8192;  F(x;&#957;, &#958;) = &#915;(&#957;/2, &#957;&#958;/2x) / &#915;(&#957;/2)
 174         </p>
 175 <p>
 176           The following graphs illustrate how the PDF and CDF of the inverse chi_squared
 177           distribution varies for a few values of parameters &#957; and &#958;:
 178         </p>
 179 <p>
 180           <span class="inlinemediaobject"><img src="../../../../graphs/inverse_chi_squared_pdf.svg" align="middle"></span>
 181         </p>
 182 <p>
 183           <span class="inlinemediaobject"><img src="../../../../graphs/inverse_chi_squared_cdf.svg" align="middle"></span>
 184         </p>
 185 <h5>
 186 <a name="math_toolkit.dist_ref.dists.inverse_chi_squared_dist.h0"></a>
 187           <span class="phrase"><a name="math_toolkit.dist_ref.dists.inverse_chi_squared_dist.member_functions"></a></span><a class="link" href="inverse_chi_squared_dist.html#math_toolkit.dist_ref.dists.inverse_chi_squared_dist.member_functions">Member
 188           Functions</a>
 189         </h5>
 190 <pre class="programlisting"><span class="identifier">inverse_chi_squared_distribution</span><span class="special">(</span><span class="identifier">RealType</span> <span class="identifier">df</span> <span class="special">=</span> <span class="number">1</span><span class="special">);</span> <span class="comment">// Implicitly scaled 1/df.</span>
 191 <span class="identifier">inverse_chi_squared_distribution</span><span class="special">(</span><span class="identifier">RealType</span> <span class="identifier">df</span> <span class="special">=</span> <span class="number">1</span><span class="special">,</span> <span class="identifier">RealType</span> <span class="identifier">scale</span><span class="special">);</span> <span class="comment">// Explicitly scaled.</span>
 192 </pre>
 193 <p>
 194           Constructs an inverse chi_squared distribution with &#957; degrees of freedom
 195           <span class="emphasis"><em>df</em></span>, and scale <span class="emphasis"><em>scale</em></span> with default
 196           value 1/df.
 197         </p>
 198 <p>
 199           Requires that the degrees of freedom &#957; parameter is greater than zero, otherwise
 200           calls <a class="link" href="../../error_handling.html#math_toolkit.error_handling.domain_error">domain_error</a>.
 201         </p>
 202 <pre class="programlisting"><span class="identifier">RealType</span> <span class="identifier">degrees_of_freedom</span><span class="special">()</span><span class="keyword">const</span><span class="special">;</span>
 203 </pre>
 204 <p>
 205           Returns the degrees_of_freedom &#957; parameter of this distribution.
 206         </p>
 207 <pre class="programlisting"><span class="identifier">RealType</span> <span class="identifier">scale</span><span class="special">()</span><span class="keyword">const</span><span class="special">;</span>
 208 </pre>
 209 <p>
 210           Returns the scale &#958; parameter of this distribution.
 211         </p>
 212 <h5>
 213 <a name="math_toolkit.dist_ref.dists.inverse_chi_squared_dist.h1"></a>
 214           <span class="phrase"><a name="math_toolkit.dist_ref.dists.inverse_chi_squared_dist.non_member_accessors"></a></span><a class="link" href="inverse_chi_squared_dist.html#math_toolkit.dist_ref.dists.inverse_chi_squared_dist.non_member_accessors">Non-member
 215           Accessors</a>
 216         </h5>
 217 <p>
 218           All the <a class="link" href="../nmp.html" title="Non-Member Properties">usual non-member accessor
 219           functions</a> that are generic to all distributions are supported:
 220           <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.cdf">Cumulative Distribution Function</a>,
 221           <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.pdf">Probability Density Function</a>,
 222           <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.quantile">Quantile</a>, <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.hazard">Hazard Function</a>, <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.chf">Cumulative Hazard Function</a>,
 223           <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.mean">mean</a>, <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.median">median</a>,
 224           <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.mode">mode</a>, <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.variance">variance</a>,
 225           <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.sd">standard deviation</a>,
 226           <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.skewness">skewness</a>, <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.kurtosis">kurtosis</a>, <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.kurtosis_excess">kurtosis_excess</a>,
 227           <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.range">range</a> and <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.support">support</a>.
 228         </p>
 229 <p>
 230           The domain of the random variate is [0,+&#8734;].
 231         </p>
 232 <div class="note"><table border="0" summary="Note">
 233 <tr>
 234 <td rowspan="2" align="center" valign="top" width="25"><img alt="[Note]" src="../../../../../../../doc/src/images/note.png"></td>
 235 <th align="left">Note</th>
 236 </tr>
 237 <tr><td align="left" valign="top"><p>
 238             Unlike some definitions, this implementation supports a random variate
 239             equal to zero as a special case, returning zero for both pdf and cdf.
 240           </p></td></tr>
 241 </table></div>
 242 <h5>
 243 <a name="math_toolkit.dist_ref.dists.inverse_chi_squared_dist.h2"></a>
 244           <span class="phrase"><a name="math_toolkit.dist_ref.dists.inverse_chi_squared_dist.accuracy"></a></span><a class="link" href="inverse_chi_squared_dist.html#math_toolkit.dist_ref.dists.inverse_chi_squared_dist.accuracy">Accuracy</a>
 245         </h5>
 246 <p>
 247           The inverse gamma distribution is implemented in terms of the incomplete
 248           gamma functions like the <a class="link" href="inverse_gamma_dist.html" title="Inverse Gamma Distribution">Inverse
 249           Gamma Distribution</a> that use <a class="link" href="../../sf_gamma/igamma.html" title="Incomplete Gamma Functions">gamma_p</a>
 250           and <a class="link" href="../../sf_gamma/igamma.html" title="Incomplete Gamma Functions">gamma_q</a> and their
 251           inverses <a class="link" href="../../sf_gamma/igamma_inv.html" title="Incomplete Gamma Function Inverses">gamma_p_inv</a>
 252           and <a class="link" href="../../sf_gamma/igamma_inv.html" title="Incomplete Gamma Function Inverses">gamma_q_inv</a>:
 253           refer to the accuracy data for those functions for more information. But
 254           in general, gamma (and thus inverse gamma) results are often accurate to
 255           a few epsilon, &gt;14 decimal digits accuracy for 64-bit double. unless
 256           iteration is involved, as for the estimation of degrees of freedom.
 257         </p>
 258 <h5>
 259 <a name="math_toolkit.dist_ref.dists.inverse_chi_squared_dist.h3"></a>
 260           <span class="phrase"><a name="math_toolkit.dist_ref.dists.inverse_chi_squared_dist.implementation"></a></span><a class="link" href="inverse_chi_squared_dist.html#math_toolkit.dist_ref.dists.inverse_chi_squared_dist.implementation">Implementation</a>
 261         </h5>
 262 <p>
 263           In the following table &#957; is the degrees of freedom parameter and &#958; is the scale
 264           parameter of the distribution, <span class="emphasis"><em>x</em></span> is the random variate,
 265           <span class="emphasis"><em>p</em></span> is the probability and <span class="emphasis"><em>q = 1-p</em></span>
 266           its complement. Parameters &#945; for shape and &#946; for scale are used for the inverse
 267           gamma function: &#945; = &#957;/2 and &#946; = &#957; * &#958;/2.
 268         </p>
 269 <div class="informaltable"><table class="table">
 270 <colgroup>
 271 <col>
 272 <col>
 273 </colgroup>
 274 <thead><tr>
 275 <th>
 276                   <p>
 277                     Function
 278                   </p>
 279                 </th>
 280 <th>
 281                   <p>
 282                     Implementation Notes
 283                   </p>
 284                 </th>
 285 </tr></thead>
 286 <tbody>
 287 <tr>
 288 <td>
 289                   <p>
 290                     pdf
 291                   </p>
 292                 </td>
 293 <td>
 294                   <p>
 295                     Using the relation: pdf = <a class="link" href="../../sf_gamma/gamma_derivatives.html" title="Derivative of the Incomplete Gamma Function">gamma_p_derivative</a>(&#945;,
 296                     &#946;/ x, &#946;) / x * x
 297                   </p>
 298                 </td>
 299 </tr>
 300 <tr>
 301 <td>
 302                   <p>
 303                     cdf
 304                   </p>
 305                 </td>
 306 <td>
 307                   <p>
 308                     Using the relation: p = <a class="link" href="../../sf_gamma/igamma.html" title="Incomplete Gamma Functions">gamma_q</a>(&#945;,
 309                     &#946; / x)
 310                   </p>
 311                 </td>
 312 </tr>
 313 <tr>
 314 <td>
 315                   <p>
 316                     cdf complement
 317                   </p>
 318                 </td>
 319 <td>
 320                   <p>
 321                     Using the relation: q = <a class="link" href="../../sf_gamma/igamma.html" title="Incomplete Gamma Functions">gamma_p</a>(&#945;,
 322                     &#946; / x)
 323                   </p>
 324                 </td>
 325 </tr>
 326 <tr>
 327 <td>
 328                   <p>
 329                     quantile
 330                   </p>
 331                 </td>
 332 <td>
 333                   <p>
 334                     Using the relation: x = &#946; &#160;/ <a class="link" href="../../sf_gamma/igamma_inv.html" title="Incomplete Gamma Function Inverses">gamma_q_inv</a>(&#945;,
 335                     p)
 336                   </p>
 337                 </td>
 338 </tr>
 339 <tr>
 340 <td>
 341                   <p>
 342                     quantile from the complement
 343                   </p>
 344                 </td>
 345 <td>
 346                   <p>
 347                     Using the relation: x = &#945; &#160;/ <a class="link" href="../../sf_gamma/igamma_inv.html" title="Incomplete Gamma Function Inverses">gamma_p_inv</a>(&#945;,
 348                     q)
 349                   </p>
 350                 </td>
 351 </tr>
 352 <tr>
 353 <td>
 354                   <p>
 355                     mode
 356                   </p>
 357                 </td>
 358 <td>
 359                   <p>
 360                     &#957; * &#958; / (&#957; + 2)
 361                   </p>
 362                 </td>
 363 </tr>
 364 <tr>
 365 <td>
 366                   <p>
 367                     median
 368                   </p>
 369                 </td>
 370 <td>
 371                   <p>
 372                     no closed form analytic equation is known, but is evaluated as
 373                     quantile(0.5)
 374                   </p>
 375                 </td>
 376 </tr>
 377 <tr>
 378 <td>
 379                   <p>
 380                     mean
 381                   </p>
 382                 </td>
 383 <td>
 384                   <p>
 385                     &#957;&#958; / (&#957; - 2) for &#957; &gt; 2, else a <a class="link" href="../../error_handling.html#math_toolkit.error_handling.domain_error">domain_error</a>
 386                   </p>
 387                 </td>
 388 </tr>
 389 <tr>
 390 <td>
 391                   <p>
 392                     variance
 393                   </p>
 394                 </td>
 395 <td>
 396                   <p>
 397                     2 &#957;&#178; &#958;&#178; / ((&#957; -2)&#178; (&#957; -4)) for &#957; &gt;4, else a <a class="link" href="../../error_handling.html#math_toolkit.error_handling.domain_error">domain_error</a>
 398                   </p>
 399                 </td>
 400 </tr>
 401 <tr>
 402 <td>
 403                   <p>
 404                     skewness
 405                   </p>
 406                 </td>
 407 <td>
 408                   <p>
 409                     4 &#8730;2 &#8730;(&#957;-4) /(&#957;-6) for &#957; &gt;6, else a <a class="link" href="../../error_handling.html#math_toolkit.error_handling.domain_error">domain_error</a>
 410                   </p>
 411                 </td>
 412 </tr>
 413 <tr>
 414 <td>
 415                   <p>
 416                     kurtosis_excess
 417                   </p>
 418                 </td>
 419 <td>
 420                   <p>
 421                     12 * (5&#957; - 22) / ((&#957; - 6) * (&#957; - 8)) for &#957; &gt;8, else a <a class="link" href="../../error_handling.html#math_toolkit.error_handling.domain_error">domain_error</a>
 422                   </p>
 423                 </td>
 424 </tr>
 425 <tr>
 426 <td>
 427                   <p>
 428                     kurtosis
 429                   </p>
 430                 </td>
 431 <td>
 432                   <p>
 433                     3 + 12 * (5&#957; - 22) / ((&#957; - 6) * (&#957;-8)) for &#957; &gt;8, else a <a class="link" href="../../error_handling.html#math_toolkit.error_handling.domain_error">domain_error</a>
 434                   </p>
 435                 </td>
 436 </tr>
 437 </tbody>
 438 </table></div>
 439 <h5>
 440 <a name="math_toolkit.dist_ref.dists.inverse_chi_squared_dist.h4"></a>
 441           <span class="phrase"><a name="math_toolkit.dist_ref.dists.inverse_chi_squared_dist.references"></a></span><a class="link" href="inverse_chi_squared_dist.html#math_toolkit.dist_ref.dists.inverse_chi_squared_dist.references">References</a>
 442         </h5>
 443 <div class="orderedlist"><ol class="orderedlist" type="1">
 444 <li class="listitem">
 445               Bayesian Data Analysis, Andrew Gelman, John B. Carlin, Hal S. Stern,
 446               Donald B. Rubin, ISBN-13: 978-1584883883, Chapman &amp; Hall; 2 edition
 447               (29 July 2003).
 448             </li>
 449 <li class="listitem">
 450               Bayesian Computation with R, Jim Albert, ISBN-13: 978-0387922973, Springer;
 451               2nd ed. edition (10 Jun 2009)
 452             </li>
 453 </ol></div>
 454 </div>
 455 <table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr>
 456 <td align="left"></td>
 457 <td align="right"><div class="copyright-footer">Copyright &#169; 2006-2010, 2012-2014 Nikhar Agrawal,
 458       Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos, Hubert
 459       Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Johan R&#229;de, Gautam Sewani,
 460       Benjamin Sobotta, Thijs van den Berg, Daryle Walker and Xiaogang Zhang<p>
 461         Distributed under the Boost Software License, Version 1.0. (See accompanying
 462         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>)
 463       </p>
 464 </div></td>
 465 </tr></table>
 466 <hr>
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