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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.nc_beta_dist"></a><a class="link" href="nc_beta_dist.html" title="Noncentral Beta Distribution">Noncentral
  28         Beta 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">non_central_beta</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">non_central_beta_distribution</span><span class="special">;</span>
  36
  37 <span class="keyword">typedef</span> <span class="identifier">non_central_beta_distribution</span><span class="special">&lt;&gt;</span> <span class="identifier">non_central_beta</span><span class="special">;</span>
  38
  39 <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">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>
  40 <span class="keyword">class</span> <span class="identifier">non_central_beta_distribution</span>
  41 <span class="special">{</span>
  42 <span class="keyword">public</span><span class="special">:</span>
  43    <span class="keyword">typedef</span> <span class="identifier">RealType</span>  <span class="identifier">value_type</span><span class="special">;</span>
  44    <span class="keyword">typedef</span> <span class="identifier">Policy</span>    <span class="identifier">policy_type</span><span class="special">;</span>
  45
  46    <span class="comment">// Constructor:</span>
  47    <span class="identifier">non_central_beta_distribution</span><span class="special">(</span><span class="identifier">RealType</span> <span class="identifier">alpha</span><span class="special">,</span> <span class="identifier">RealType</span> <span class="identifier">beta</span><span class="special">,</span> <span class="identifier">RealType</span> <span class="identifier">lambda</span><span class="special">);</span>
  48
  49    <span class="comment">// Accessor to shape parameters:</span>
  50    <span class="identifier">RealType</span> <span class="identifier">alpha</span><span class="special">()</span><span class="keyword">const</span><span class="special">;</span>
  51    <span class="identifier">RealType</span> <span class="identifier">beta</span><span class="special">()</span><span class="keyword">const</span><span class="special">;</span>
  52
  53    <span class="comment">// Accessor to non-centrality parameter lambda:</span>
  54    <span class="identifier">RealType</span> <span class="identifier">non_centrality</span><span class="special">()</span><span class="keyword">const</span><span class="special">;</span>
  55 <span class="special">};</span>
  56
  57 <span class="special">}}</span> <span class="comment">// namespaces</span>
  58 </pre>
  59 <p>
  60           The noncentral beta distribution is a generalization of the <a class="link" href="beta_dist.html" title="Beta Distribution">Beta
  61           Distribution</a>.
  62         </p>
  63 <p>
  64           It is defined as the ratio X = &#967;<sub>m</sub><sup>2</sup>(&#955;) / (&#967;<sub>m</sub><sup>2</sup>(&#955;) + &#967;<sub>n</sub><sup>2</sup>) where &#967;<sub>m</sub><sup>2</sup>(&#955;) is a noncentral
  65           &#967;<sup>2</sup>
  66 random variable with <span class="emphasis"><em>m</em></span> degrees of freedom, and &#967;<sub>n</sub><sup>2</sup>
  67 is
  68           a central &#967;<sup>2</sup> random variable with <span class="emphasis"><em>n</em></span> degrees of freedom.
  69         </p>
  70 <p>
  71           This gives a PDF that can be expressed as a Poisson mixture of beta distribution
  72           PDFs:
  73         </p>
  74 <p>
  75           <span class="inlinemediaobject"><img src="../../../../equations/nc_beta_ref1.svg"></span>
  76         </p>
  77 <p>
  78           where P(i;&#955;/2) is the discrete Poisson probablity at <span class="emphasis"><em>i</em></span>,
  79           with mean &#955;/2, and I<sub>x</sub><sup>'</sup>(&#945;, &#946;) is the derivative of the incomplete beta function.
  80           This leads to the usual form of the CDF as:
  81         </p>
  82 <p>
  83           <span class="inlinemediaobject"><img src="../../../../equations/nc_beta_ref2.svg"></span>
  84         </p>
  85 <p>
  86           The following graph illustrates how the distribution changes for different
  87           values of &#955;:
  88         </p>
  89 <p>
  90           <span class="inlinemediaobject"><img src="../../../../graphs/nc_beta_pdf.svg" align="middle"></span>
  91         </p>
  92 <h5>
  93 <a name="math_toolkit.dist_ref.dists.nc_beta_dist.h0"></a>
  94           <span class="phrase"><a name="math_toolkit.dist_ref.dists.nc_beta_dist.member_functions"></a></span><a class="link" href="nc_beta_dist.html#math_toolkit.dist_ref.dists.nc_beta_dist.member_functions">Member
  95           Functions</a>
  96         </h5>
  97 <pre class="programlisting"><span class="identifier">non_central_beta_distribution</span><span class="special">(</span><span class="identifier">RealType</span> <span class="identifier">a</span><span class="special">,</span> <span class="identifier">RealType</span> <span class="identifier">b</span><span class="special">,</span> <span class="identifier">RealType</span> <span class="identifier">lambda</span><span class="special">);</span>
  98 </pre>
  99 <p>
 100           Constructs a noncentral beta distribution with shape parameters <span class="emphasis"><em>a</em></span>
 101           and <span class="emphasis"><em>b</em></span> and non-centrality parameter <span class="emphasis"><em>lambda</em></span>.
 102         </p>
 103 <p>
 104           Requires a &gt; 0, b &gt; 0 and lambda &gt;= 0, otherwise calls <a class="link" href="../../error_handling.html#math_toolkit.error_handling.domain_error">domain_error</a>.
 105         </p>
 106 <pre class="programlisting"><span class="identifier">RealType</span> <span class="identifier">alpha</span><span class="special">()</span><span class="keyword">const</span><span class="special">;</span>
 107 </pre>
 108 <p>
 109           Returns the parameter <span class="emphasis"><em>a</em></span> from which this object was
 110           constructed.
 111         </p>
 112 <pre class="programlisting"><span class="identifier">RealType</span> <span class="identifier">beta</span><span class="special">()</span><span class="keyword">const</span><span class="special">;</span>
 113 </pre>
 114 <p>
 115           Returns the parameter <span class="emphasis"><em>b</em></span> from which this object was
 116           constructed.
 117         </p>
 118 <pre class="programlisting"><span class="identifier">RealType</span> <span class="identifier">non_centrality</span><span class="special">()</span><span class="keyword">const</span><span class="special">;</span>
 119 </pre>
 120 <p>
 121           Returns the parameter <span class="emphasis"><em>lambda</em></span> from which this object
 122           was constructed.
 123         </p>
 124 <h5>
 125 <a name="math_toolkit.dist_ref.dists.nc_beta_dist.h1"></a>
 126           <span class="phrase"><a name="math_toolkit.dist_ref.dists.nc_beta_dist.non_member_accessors"></a></span><a class="link" href="nc_beta_dist.html#math_toolkit.dist_ref.dists.nc_beta_dist.non_member_accessors">Non-member
 127           Accessors</a>
 128         </h5>
 129 <p>
 130           Most of the <a class="link" href="../nmp.html" title="Non-Member Properties">usual non-member
 131           accessor functions</a> are supported: <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.cdf">Cumulative
 132           Distribution Function</a>, <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.pdf">Probability
 133           Density Function</a>, <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.quantile">Quantile</a>,
 134           <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.variance">variance</a>,
 135           <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.sd">standard deviation</a>,
 136           <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.median">median</a>, <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.mode">mode</a>,
 137           <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.hazard">Hazard Function</a>,
 138           <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.chf">Cumulative Hazard Function</a>,
 139           <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>.
 140         </p>
 141 <p>
 142           Mean and variance are implemented using hypergeometric pfq functions and
 143           relations given in <a href="http://reference.wolfram.com/mathematica/ref/NoncentralBetaDistribution.html" target="_top">Wolfram
 144           Noncentral Beta Distribution</a>.
 145         </p>
 146 <p>
 147           However, the following are not currently implemented: <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.skewness">skewness</a>,
 148           <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.kurtosis">kurtosis</a> and
 149           <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.kurtosis_excess">kurtosis_excess</a>.
 150         </p>
 151 <p>
 152           The domain of the random variable is [0, 1].
 153         </p>
 154 <h5>
 155 <a name="math_toolkit.dist_ref.dists.nc_beta_dist.h2"></a>
 156           <span class="phrase"><a name="math_toolkit.dist_ref.dists.nc_beta_dist.accuracy"></a></span><a class="link" href="nc_beta_dist.html#math_toolkit.dist_ref.dists.nc_beta_dist.accuracy">Accuracy</a>
 157         </h5>
 158 <p>
 159           The following table shows the peak errors (in units of <a href="http://en.wikipedia.org/wiki/Machine_epsilon" target="_top">epsilon</a>)
 160           found on various platforms with various floating point types. The failures
 161           in the comparison to the <a href="http://www.r-project.org/" target="_top">R Math
 162           library</a>, seem to be mostly in the corner cases when the probablity
 163           would be very small. Unless otherwise specified any floating-point type
 164           that is narrower than the one shown will have <a class="link" href="../../relative_error.html#math_toolkit.relative_error.zero_error">effectively
 165           zero error</a>.
 166         </p>
 167 <div class="table">
 168 <a name="math_toolkit.dist_ref.dists.nc_beta_dist.table_non_central_beta_CDF"></a><p class="title"><b>Table&#160;5.4.&#160;Error rates for non central beta CDF</b></p>
 169 <div class="table-contents"><table class="table" summary="Error rates for non central beta CDF">
 170 <colgroup>
 171 <col>
 172 <col>
 173 <col>
 174 <col>
 175 <col>
 176 </colgroup>
 177 <thead><tr>
 178 <th>
 179                 </th>
 180 <th>
 181                   <p>
 182                     Microsoft Visual C++ version 12.0<br> Win32<br> double
 183                   </p>
 184                 </th>
 185 <th>
 186                   <p>
 187                     GNU C++ version 5.1.0<br> linux<br> double
 188                   </p>
 189                 </th>
 190 <th>
 191                   <p>
 192                     GNU C++ version 5.1.0<br> linux<br> long double
 193                   </p>
 194                 </th>
 195 <th>
 196                   <p>
 197                     Sun compiler version 0x5130<br> Sun Solaris<br> long double
 198                   </p>
 199                 </th>
 200 </tr></thead>
 201 <tbody>
 202 <tr>
 203 <td>
 204                   <p>
 205                     Non Central Beta, medium parameters
 206                   </p>
 207                 </td>
 208 <td>
 209                   <p>
 210                     <span class="blue">Max = 240&#949; (Mean = 31&#949;)</span>
 211                   </p>
 212                 </td>
 213 <td>
 214                   <p>
 215                     <span class="blue">Max = 0.998&#949; (Mean = 0.0659&#949;)</span><br>
 216                     <br> (<span class="emphasis"><em>Rmath 3.0.2:</em></span> <span class="red">Max
 217                     = 1.46e+26&#949; (Mean = 3.5e+24&#949;) <a class="link" href="../../logs_and_tables/logs.html#errors_GNU_C_version_5_1_0_linux_double_non_central_beta_CDF_Rmath_3_0_2_Non_Central_Beta_medium_parameters">And
 218                     other failures.</a>)</span>
 219                   </p>
 220                 </td>
 221 <td>
 222                   <p>
 223                     <span class="blue">Max = 825&#949; (Mean = 27.4&#949;)</span>
 224                   </p>
 225                 </td>
 226 <td>
 227                   <p>
 228                     <span class="blue">Max = 832&#949; (Mean = 38.1&#949;)</span>
 229                   </p>
 230                 </td>
 231 </tr>
 232 <tr>
 233 <td>
 234                   <p>
 235                     Non Central Beta, large parameters
 236                   </p>
 237                 </td>
 238 <td>
 239                   <p>
 240                     <span class="blue">Max = 3.41e+003&#949; (Mean = 475&#949;)</span>
 241                   </p>
 242                 </td>
 243 <td>
 244                   <p>
 245                     <span class="blue">Max = 1.18&#949; (Mean = 0.175&#949;)</span><br>
 246                     <br> (<span class="emphasis"><em>Rmath 3.0.2:</em></span> <span class="red">Max
 247                     = 1.01e+36&#949; (Mean = 1.19e+35&#949;) <a class="link" href="../../logs_and_tables/logs.html#errors_GNU_C_version_5_1_0_linux_double_non_central_beta_CDF_Rmath_3_0_2_Non_Central_Beta_large_parameters">And
 248                     other failures.</a>)</span>
 249                   </p>
 250                 </td>
 251 <td>
 252                   <p>
 253                     <span class="blue">Max = 2.5e+04&#949; (Mean = 3.78e+03&#949;)</span>
 254                   </p>
 255                 </td>
 256 <td>
 257                   <p>
 258                     <span class="blue">Max = 2.57e+04&#949; (Mean = 4.43e+03&#949;)</span>
 259                   </p>
 260                 </td>
 261 </tr>
 262 </tbody>
 263 </table></div>
 264 </div>
 265 <br class="table-break"><div class="table">
 266 <a name="math_toolkit.dist_ref.dists.nc_beta_dist.table_non_central_beta_CDF_complement"></a><p class="title"><b>Table&#160;5.5.&#160;Error rates for non central beta CDF complement</b></p>
 267 <div class="table-contents"><table class="table" summary="Error rates for non central beta CDF complement">
 268 <colgroup>
 269 <col>
 270 <col>
 271 <col>
 272 <col>
 273 <col>
 274 </colgroup>
 275 <thead><tr>
 276 <th>
 277                 </th>
 278 <th>
 279                   <p>
 280                     Microsoft Visual C++ version 12.0<br> Win32<br> double
 281                   </p>
 282                 </th>
 283 <th>
 284                   <p>
 285                     GNU C++ version 5.1.0<br> linux<br> double
 286                   </p>
 287                 </th>
 288 <th>
 289                   <p>
 290                     GNU C++ version 5.1.0<br> linux<br> long double
 291                   </p>
 292                 </th>
 293 <th>
 294                   <p>
 295                     Sun compiler version 0x5130<br> Sun Solaris<br> long double
 296                   </p>
 297                 </th>
 298 </tr></thead>
 299 <tbody>
 300 <tr>
 301 <td>
 302                   <p>
 303                     Non Central Beta, medium parameters
 304                   </p>
 305                 </td>
 306 <td>
 307                   <p>
 308                     <span class="blue">Max = 619&#949; (Mean = 62.7&#949;)</span>
 309                   </p>
 310                 </td>
 311 <td>
 312                   <p>
 313                     <span class="blue">Max = 0.998&#949; (Mean = 0.0957&#949;)</span><br>
 314                     <br> (<span class="emphasis"><em>Rmath 3.0.2:</em></span> <span class="red">Max
 315                     = 7.5e+97&#949; (Mean = 1.37e+96&#949;) <a class="link" href="../../logs_and_tables/logs.html#errors_GNU_C_version_5_1_0_linux_double_non_central_beta_CDF_complement_Rmath_3_0_2_Non_Central_Beta_medium_parameters">And
 316                     other failures.</a>)</span>
 317                   </p>
 318                 </td>
 319 <td>
 320                   <p>
 321                     <span class="blue">Max = 396&#949; (Mean = 50.7&#949;)</span>
 322                   </p>
 323                 </td>
 324 <td>
 325                   <p>
 326                     <span class="blue">Max = 554&#949; (Mean = 57.3&#949;)</span>
 327                   </p>
 328                 </td>
 329 </tr>
 330 <tr>
 331 <td>
 332                   <p>
 333                     Non Central Beta, large parameters
 334                   </p>
 335                 </td>
 336 <td>
 337                   <p>
 338                     <span class="blue">Max = 8.67e+003&#949; (Mean = 1.04e+003&#949;)</span>
 339                   </p>
 340                 </td>
 341 <td>
 342                   <p>
 343                     <span class="blue">Max = 0.986&#949; (Mean = 0.188&#949;)</span><br>
 344                     <br> (<span class="emphasis"><em>Rmath 3.0.2:</em></span> <span class="red">Max
 345                     = +INF&#949; (Mean = +INF&#949;) <a class="link" href="../../logs_and_tables/logs.html#errors_GNU_C_version_5_1_0_linux_double_non_central_beta_CDF_complement_Rmath_3_0_2_Non_Central_Beta_large_parameters">And
 346                     other failures.</a>)</span>
 347                   </p>
 348                 </td>
 349 <td>
 350                   <p>
 351                     <span class="blue">Max = 6.83e+03&#949; (Mean = 993&#949;)</span>
 352                   </p>
 353                 </td>
 354 <td>
 355                   <p>
 356                     <span class="blue">Max = 3.56e+03&#949; (Mean = 704&#949;)</span>
 357                   </p>
 358                 </td>
 359 </tr>
 360 </tbody>
 361 </table></div>
 362 </div>
 363 <br class="table-break"><p>
 364           Error rates for the PDF, the complement of the CDF and for the quantile
 365           functions are broadly similar.
 366         </p>
 367 <h5>
 368 <a name="math_toolkit.dist_ref.dists.nc_beta_dist.h3"></a>
 369           <span class="phrase"><a name="math_toolkit.dist_ref.dists.nc_beta_dist.tests"></a></span><a class="link" href="nc_beta_dist.html#math_toolkit.dist_ref.dists.nc_beta_dist.tests">Tests</a>
 370         </h5>
 371 <p>
 372           There are two sets of test data used to verify this implementation: firstly
 373           we can compare with a few sample values generated by the <a href="http://www.r-project.org/" target="_top">R
 374           library</a>. Secondly, we have tables of test data, computed with this
 375           implementation and using interval arithmetic - this data should be accurate
 376           to at least 50 decimal digits - and is the used for our accuracy tests.
 377         </p>
 378 <h5>
 379 <a name="math_toolkit.dist_ref.dists.nc_beta_dist.h4"></a>
 380           <span class="phrase"><a name="math_toolkit.dist_ref.dists.nc_beta_dist.implementation"></a></span><a class="link" href="nc_beta_dist.html#math_toolkit.dist_ref.dists.nc_beta_dist.implementation">Implementation</a>
 381         </h5>
 382 <p>
 383           The CDF and its complement are evaluated as follows:
 384         </p>
 385 <p>
 386           First we determine which of the two values (the CDF or its complement)
 387           is likely to be the smaller, the crossover point is taken to be the mean
 388           of the distribution: for this we use the approximation due to: R. Chattamvelli
 389           and R. Shanmugam, "Algorithm AS 310: Computing the Non-Central Beta
 390           Distribution Function", Applied Statistics, Vol. 46, No. 1. (1997),
 391           pp. 146-156.
 392         </p>
 393 <p>
 394           <span class="inlinemediaobject"><img src="../../../../equations/nc_beta_ref3.svg"></span>
 395         </p>
 396 <p>
 397           Then either the CDF or its complement is computed using the relations:
 398         </p>
 399 <p>
 400           <span class="inlinemediaobject"><img src="../../../../equations/nc_beta_ref4.svg"></span>
 401         </p>
 402 <p>
 403           The summation is performed by starting at i = &#955;/2, and then recursing in
 404           both directions, using the usual recurrence relations for the Poisson PDF
 405           and incomplete beta functions. This is the "Method 2" described
 406           by:
 407         </p>
 408 <p>
 409           Denise Benton and K. Krishnamoorthy, "Computing discrete mixtures
 410           of continuous distributions: noncentral chisquare, noncentral t and the
 411           distribution of the square of the sample multiple correlation coefficient",
 412           Computational Statistics &amp; Data Analysis 43 (2003) 249-267.
 413         </p>
 414 <p>
 415           Specific applications of the above formulae to the noncentral beta distribution
 416           can be found in:
 417         </p>
 418 <p>
 419           Russell V. Lenth, "Algorithm AS 226: Computing Noncentral Beta Probabilities",
 420           Applied Statistics, Vol. 36, No. 2. (1987), pp. 241-244.
 421         </p>
 422 <p>
 423           H. Frick, "Algorithm AS R84: A Remark on Algorithm AS 226: Computing
 424           Non-Central Beta Probabilities", Applied Statistics, Vol. 39, No.
 425           2. (1990), pp. 311-312.
 426         </p>
 427 <p>
 428           Ming Long Lam, "Remark AS R95: A Remark on Algorithm AS 226: Computing
 429           Non-Central Beta Probabilities", Applied Statistics, Vol. 44, No.
 430           4. (1995), pp. 551-552.
 431         </p>
 432 <p>
 433           Harry O. Posten, "An Effective Algorithm for the Noncentral Beta Distribution
 434           Function", The American Statistician, Vol. 47, No. 2. (May, 1993),
 435           pp. 129-131.
 436         </p>
 437 <p>
 438           R. Chattamvelli, "A Note on the Noncentral Beta Distribution Function",
 439           The American Statistician, Vol. 49, No. 2. (May, 1995), pp. 231-234.
 440         </p>
 441 <p>
 442           Of these, the Posten reference provides the most complete overview, and
 443           includes the modification starting iteration at &#955;/2.
 444         </p>
 445 <p>
 446           The main difference between this implementation and the above references
 447           is the direct computation of the complement when most efficient to do so,
 448           and the accumulation of the sum to -1 rather than subtracting the result
 449           from 1 at the end: this can substantially reduce the number of iterations
 450           required when the result is near 1.
 451         </p>
 452 <p>
 453           The PDF is computed using the methodology of Benton and Krishnamoorthy
 454           and the relation:
 455         </p>
 456 <p>
 457           <span class="inlinemediaobject"><img src="../../../../equations/nc_beta_ref1.svg"></span>
 458         </p>
 459 <p>
 460           Quantiles are computed using a specially modified version of <a class="link" href="../../roots/roots_noderiv/bracket_solve.html" title="Bracket and Solve Root">bracket
 461           and solve</a>, starting the search for the root at the mean of the distribution.
 462           (A Cornish-Fisher type expansion was also tried, but while this gets quite
 463           close to the root in many cases, when it is wrong it tends to introduce
 464           quite pathological behaviour: more investigation in this area is probably
 465           warranted).
 466         </p>
 467 </div>
 468 <table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr>
 469 <td align="left"></td>
 470 <td align="right"><div class="copyright-footer">Copyright &#169; 2006-2010, 2012-2014 Nikhar Agrawal,
 471       Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos, Hubert
 472       Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Johan R&#229;de, Gautam Sewani,
 473       Benjamin Sobotta, Thijs van den Berg, Daryle Walker and Xiaogang Zhang<p>
 474         Distributed under the Boost Software License, Version 1.0. (See accompanying
 475         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>)
 476       </p>
 477 </div></td>
 478 </tr></table>
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