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  26 <div class="titlepage"><div><div><h3 class="title">
  27 <a name="math_toolkit.ellint.ellint_intro"></a><a class="link" href="ellint_intro.html" title="Elliptic Integral Overview">Elliptic Integral Overview</a>
  28 </h3></div></div></div>
  29 <p>
  30         The main reference for the elliptic integrals is:
  31       </p>
  32 <div class="blockquote"><blockquote class="blockquote"><p>
  33           M. Abramowitz and I. A. Stegun (Eds.) (1964) Handbook of Mathematical Functions
  34           with Formulas, Graphs, and Mathematical Tables, National Bureau of Standards
  35           Applied Mathematics Series, U.S. Government Printing Office, Washington,
  36           D.C.
  37         </p></blockquote></div>
  38 <p>
  39         and its recently revised version <a href="http://dlmf.nist.gov/" target="_top">NIST
  40         Digital Library of Mathematical Functions (DMLF)</a>, in particular
  41       </p>
  42 <div class="blockquote"><blockquote class="blockquote"><p>
  43           <a href="https://dlmf.nist.gov/19" target="_top">Elliptic Integrals, B. C. Carlson</a>
  44         </p></blockquote></div>
  45 <p>
  46         Mathworld also contain a lot of useful background information:
  47       </p>
  48 <div class="blockquote"><blockquote class="blockquote"><p>
  49           <a href="http://mathworld.wolfram.com/EllipticIntegral.html" target="_top">Weisstein,
  50           Eric W. "Elliptic Integral." From MathWorld--A Wolfram Web Resource.</a>
  51         </p></blockquote></div>
  52 <p>
  53         As does <a href="http://en.wikipedia.org/wiki/Elliptic_integral" target="_top">Wikipedia
  54         Elliptic integral</a>.
  55       </p>
  56 <h5>
  57 <a name="math_toolkit.ellint.ellint_intro.h0"></a>
  58         <span class="phrase"><a name="math_toolkit.ellint.ellint_intro.notation"></a></span><a class="link" href="ellint_intro.html#math_toolkit.ellint.ellint_intro.notation">Notation</a>
  59       </h5>
  60 <p>
  61         All variables are real numbers unless otherwise noted.
  62       </p>
  63 <h5>
  64 <a name="math_toolkit.ellint.ellint_intro.h1"></a>
  65         <span class="phrase"><a name="math_toolkit.ellint.ellint_intro.definition"></a></span><a class="link" href="ellint_intro.html#math_toolkit.ellint.ellint_intro.definition">Definition</a>
  66       </h5>
  67 <div class="blockquote"><blockquote class="blockquote"><p>
  68           <span class="inlinemediaobject"><img src="../../../equations/ellint1.svg"></span>
  69
  70         </p></blockquote></div>
  71 <p>
  72         is called elliptic integral if <span class="emphasis"><em>R(t, s)</em></span> is a rational
  73         function of <span class="emphasis"><em>t</em></span> and <span class="emphasis"><em>s</em></span>, and <span class="emphasis"><em>s<sup>2</sup></em></span>
  74         is a cubic or quartic polynomial in <span class="emphasis"><em>t</em></span>.
  75       </p>
  76 <p>
  77         Elliptic integrals generally cannot be expressed in terms of elementary functions.
  78         However, Legendre showed that all elliptic integrals can be reduced to the
  79         following three canonical forms:
  80       </p>
  81 <p>
  82         Elliptic Integral of the First Kind (Legendre form)
  83       </p>
  84 <div class="blockquote"><blockquote class="blockquote"><p>
  85           <span class="inlinemediaobject"><img src="../../../equations/ellint2.svg"></span>
  86
  87         </p></blockquote></div>
  88 <p>
  89         Elliptic Integral of the Second Kind (Legendre form)
  90       </p>
  91 <div class="blockquote"><blockquote class="blockquote"><p>
  92           <span class="inlinemediaobject"><img src="../../../equations/ellint3.svg"></span>
  93
  94         </p></blockquote></div>
  95 <p>
  96         Elliptic Integral of the Third Kind (Legendre form)
  97       </p>
  98 <div class="blockquote"><blockquote class="blockquote"><p>
  99           <span class="inlinemediaobject"><img src="../../../equations/ellint4.svg"></span>
 100
 101         </p></blockquote></div>
 102 <p>
 103         where
 104       </p>
 105 <div class="blockquote"><blockquote class="blockquote"><p>
 106           <span class="inlinemediaobject"><img src="../../../equations/ellint5.svg"></span>
 107
 108         </p></blockquote></div>
 109 <div class="note"><table border="0" summary="Note">
 110 <tr>
 111 <td rowspan="2" align="center" valign="top" width="25"><img alt="[Note]" src="../../../../../../doc/src/images/note.png"></td>
 112 <th align="left">Note</th>
 113 </tr>
 114 <tr><td align="left" valign="top">
 115 <p>
 116           <span class="emphasis"><em>&#966;</em></span> is called the amplitude.
 117         </p>
 118 <p>
 119           <span class="emphasis"><em>k</em></span> is called the elliptic modulus or eccentricity.
 120         </p>
 121 <p>
 122           <span class="emphasis"><em>&#945;</em></span> is called the modular angle.
 123         </p>
 124 <p>
 125           <span class="emphasis"><em>n</em></span> is called the characteristic.
 126         </p>
 127 </td></tr>
 128 </table></div>
 129 <div class="caution"><table border="0" summary="Caution">
 130 <tr>
 131 <td rowspan="2" align="center" valign="top" width="25"><img alt="[Caution]" src="../../../../../../doc/src/images/caution.png"></td>
 132 <th align="left">Caution</th>
 133 </tr>
 134 <tr><td align="left" valign="top">
 135 <p>
 136           Perhaps more than any other special functions the elliptic integrals are
 137           expressed in a variety of different ways. In particular, the final parameter
 138           <span class="emphasis"><em>k</em></span> (the modulus) may be expressed using a modular angle
 139           &#945;, or a parameter <span class="emphasis"><em>m</em></span>. These are related by:
 140         </p>
 141 <div class="blockquote"><blockquote class="blockquote"><p>
 142             <span class="serif_italic">k = sin &#8201;&#945;</span>
 143           </p></blockquote></div>
 144 <div class="blockquote"><blockquote class="blockquote"><p>
 145             <span class="serif_italic">m = k<sup>2</sup> = sin<sup>2</sup>&#945;</span>
 146           </p></blockquote></div>
 147 <p>
 148           So that the integral of the third kind (for example) may be expressed as
 149           either:
 150         </p>
 151 <div class="blockquote"><blockquote class="blockquote"><p>
 152             <span class="serif_italic">&#928;(n, &#966;, k)</span>
 153           </p></blockquote></div>
 154 <div class="blockquote"><blockquote class="blockquote"><p>
 155             <span class="serif_italic">&#928;(n, &#966; \ &#945;)</span>
 156           </p></blockquote></div>
 157 <div class="blockquote"><blockquote class="blockquote"><p>
 158             <span class="serif_italic">&#928;(n, &#966; | m)</span>
 159           </p></blockquote></div>
 160 <p>
 161           To further complicate matters, some texts refer to the <span class="emphasis"><em>complement
 162           of the parameter m</em></span>, or 1 - m, where:
 163         </p>
 164 <div class="blockquote"><blockquote class="blockquote"><p>
 165             <span class="serif_italic">1 - m = 1 - k<sup>2</sup> = cos<sup>2</sup>&#945;</span>
 166           </p></blockquote></div>
 167 <p>
 168           This implementation uses <span class="emphasis"><em>k</em></span> throughout: this matches
 169           the requirements of the <a href="http://www.open-std.org/jtc1/sc22/wg21/docs/papers/2005/n1836.pdf" target="_top">Technical
 170           Report on C++ Library Extensions</a>.<br>
 171         </p>
 172 <p>
 173           So you should be extra careful when using these functions!
 174         </p>
 175 </td></tr>
 176 </table></div>
 177 <div class="warning"><table border="0" summary="Warning">
 178 <tr>
 179 <td rowspan="2" align="center" valign="top" width="25"><img alt="[Warning]" src="../../../../../../doc/src/images/warning.png"></td>
 180 <th align="left">Warning</th>
 181 </tr>
 182 <tr><td align="left" valign="top"><p>
 183           Boost.Math order of arguments differs from other implementations: <span class="emphasis"><em>k</em></span>
 184           is always the <span class="bold"><strong>first</strong></span> argument.
 185         </p></td></tr>
 186 </table></div>
 187 <p>
 188         A simple example comparing use of <a href="http://www.wolframalpha.com/" target="_top">Wolfram
 189         Alpha</a> with Boost.Math (including much higher precision using <a href="../../../../../../libs/multiprecision/doc/html/index.html" target="_top">Boost.Multiprecision</a>)
 190         is <a href="../../../../example/jacobi_zeta_example.cpp" target="_top">jacobi_zeta_example.cpp</a>.
 191       </p>
 192 <p>
 193         When <span class="emphasis"><em>&#966;</em></span> = <span class="emphasis"><em>&#960;</em></span> / 2, the elliptic integrals
 194         are called <span class="emphasis"><em>complete</em></span>.
 195       </p>
 196 <p>
 197         Complete Elliptic Integral of the First Kind (Legendre form)
 198       </p>
 199 <div class="blockquote"><blockquote class="blockquote"><p>
 200           <span class="inlinemediaobject"><img src="../../../equations/ellint6.svg"></span>
 201
 202         </p></blockquote></div>
 203 <p>
 204         Complete Elliptic Integral of the Second Kind (Legendre form)
 205       </p>
 206 <div class="blockquote"><blockquote class="blockquote"><p>
 207           <span class="inlinemediaobject"><img src="../../../equations/ellint7.svg"></span>
 208
 209         </p></blockquote></div>
 210 <p>
 211         Complete Elliptic Integral of the Third Kind (Legendre form)
 212       </p>
 213 <div class="blockquote"><blockquote class="blockquote"><p>
 214           <span class="inlinemediaobject"><img src="../../../equations/ellint8.svg"></span>
 215
 216         </p></blockquote></div>
 217 <p>
 218         Legendre also defined a fourth integral /D(&#966;,k)/ which is a combination of
 219         the other three:
 220       </p>
 221 <div class="blockquote"><blockquote class="blockquote"><p>
 222           <span class="inlinemediaobject"><img src="../../../equations/ellint_d.svg"></span>
 223
 224         </p></blockquote></div>
 225 <p>
 226         Like the other Legendre integrals this comes in both complete and incomplete
 227         forms.
 228       </p>
 229 <h5>
 230 <a name="math_toolkit.ellint.ellint_intro.h2"></a>
 231         <span class="phrase"><a name="math_toolkit.ellint.ellint_intro.carlson_elliptic_integrals"></a></span><a class="link" href="ellint_intro.html#math_toolkit.ellint.ellint_intro.carlson_elliptic_integrals">Carlson
 232         Elliptic Integrals</a>
 233       </h5>
 234 <p>
 235         Carlson [<a class="link" href="ellint_intro.html#ellint_ref_carlson77">Carlson77</a>] [<a class="link" href="ellint_intro.html#ellint_ref_carlson78">Carlson78</a>]
 236         gives an alternative definition of elliptic integral's canonical forms:
 237       </p>
 238 <p>
 239         Carlson's Elliptic Integral of the First Kind
 240       </p>
 241 <div class="blockquote"><blockquote class="blockquote"><p>
 242           <span class="inlinemediaobject"><img src="../../../equations/ellint9.svg"></span>
 243
 244         </p></blockquote></div>
 245 <p>
 246         where <span class="emphasis"><em>x</em></span>, <span class="emphasis"><em>y</em></span>, <span class="emphasis"><em>z</em></span>
 247         are nonnegative and at most one of them may be zero.
 248       </p>
 249 <p>
 250         Carlson's Elliptic Integral of the Second Kind
 251       </p>
 252 <div class="blockquote"><blockquote class="blockquote"><p>
 253           <span class="inlinemediaobject"><img src="../../../equations/ellint10.svg"></span>
 254
 255         </p></blockquote></div>
 256 <p>
 257         where <span class="emphasis"><em>x</em></span>, <span class="emphasis"><em>y</em></span> are nonnegative, at
 258         most one of them may be zero, and <span class="emphasis"><em>z</em></span> must be positive.
 259       </p>
 260 <p>
 261         Carlson's Elliptic Integral of the Third Kind
 262       </p>
 263 <div class="blockquote"><blockquote class="blockquote"><p>
 264           <span class="inlinemediaobject"><img src="../../../equations/ellint11.svg"></span>
 265
 266         </p></blockquote></div>
 267 <p>
 268         where <span class="emphasis"><em>x</em></span>, <span class="emphasis"><em>y</em></span>, <span class="emphasis"><em>z</em></span>
 269         are nonnegative, at most one of them may be zero, and <span class="emphasis"><em>p</em></span>
 270         must be nonzero.
 271       </p>
 272 <p>
 273         Carlson's Degenerate Elliptic Integral
 274       </p>
 275 <div class="blockquote"><blockquote class="blockquote"><p>
 276           <span class="inlinemediaobject"><img src="../../../equations/ellint12.svg"></span>
 277
 278         </p></blockquote></div>
 279 <p>
 280         where <span class="emphasis"><em>x</em></span> is nonnegative and <span class="emphasis"><em>y</em></span> is
 281         nonzero.
 282       </p>
 283 <div class="note"><table border="0" summary="Note">
 284 <tr>
 285 <td rowspan="2" align="center" valign="top" width="25"><img alt="[Note]" src="../../../../../../doc/src/images/note.png"></td>
 286 <th align="left">Note</th>
 287 </tr>
 288 <tr><td align="left" valign="top">
 289 <p>
 290           <span class="emphasis"><em>R<sub>C</sub>(x, y) = R<sub>F</sub>(x, y, y)</em></span>
 291         </p>
 292 <p>
 293           <span class="emphasis"><em>R<sub>D</sub>(x, y, z) = R<sub>J</sub>(x, y, z, z)</em></span>
 294         </p>
 295 </td></tr>
 296 </table></div>
 297 <p>
 298         Carlson's Symmetric Integral
 299       </p>
 300 <div class="blockquote"><blockquote class="blockquote"><p>
 301           <span class="inlinemediaobject"><img src="../../../equations/ellint27.svg"></span>
 302
 303         </p></blockquote></div>
 304 <h5>
 305 <a name="math_toolkit.ellint.ellint_intro.h3"></a>
 306         <span class="phrase"><a name="math_toolkit.ellint.ellint_intro.duplication_theorem"></a></span><a class="link" href="ellint_intro.html#math_toolkit.ellint.ellint_intro.duplication_theorem">Duplication
 307         Theorem</a>
 308       </h5>
 309 <p>
 310         Carlson proved in [<a class="link" href="ellint_intro.html#ellint_ref_carlson78">Carlson78</a>]
 311         that
 312       </p>
 313 <div class="blockquote"><blockquote class="blockquote"><p>
 314           <span class="inlinemediaobject"><img src="../../../equations/ellint13.svg"></span>
 315
 316         </p></blockquote></div>
 317 <h5>
 318 <a name="math_toolkit.ellint.ellint_intro.h4"></a>
 319         <span class="phrase"><a name="math_toolkit.ellint.ellint_intro.carlson_s_formulas"></a></span><a class="link" href="ellint_intro.html#math_toolkit.ellint.ellint_intro.carlson_s_formulas">Carlson's
 320         Formulas</a>
 321       </h5>
 322 <p>
 323         The Legendre form and Carlson form of elliptic integrals are related by equations:
 324       </p>
 325 <div class="blockquote"><blockquote class="blockquote"><p>
 326           <span class="inlinemediaobject"><img src="../../../equations/ellint14.svg"></span>
 327
 328         </p></blockquote></div>
 329 <p>
 330         In particular,
 331       </p>
 332 <div class="blockquote"><blockquote class="blockquote"><p>
 333           <span class="inlinemediaobject"><img src="../../../equations/ellint15.svg"></span>
 334
 335         </p></blockquote></div>
 336 <h5>
 337 <a name="math_toolkit.ellint.ellint_intro.h5"></a>
 338         <span class="phrase"><a name="math_toolkit.ellint.ellint_intro.miscellaneous_elliptic_integrals"></a></span><a class="link" href="ellint_intro.html#math_toolkit.ellint.ellint_intro.miscellaneous_elliptic_integrals">Miscellaneous
 339         Elliptic Integrals</a>
 340       </h5>
 341 <p>
 342         There are two functions related to the elliptic integrals which otherwise
 343         defy categorisation, these are the Jacobi Zeta function:
 344       </p>
 345 <div class="blockquote"><blockquote class="blockquote"><p>
 346           <span class="inlinemediaobject"><img src="../../../equations/jacobi_zeta.svg"></span>
 347
 348         </p></blockquote></div>
 349 <p>
 350         and the Heuman Lambda function:
 351       </p>
 352 <div class="blockquote"><blockquote class="blockquote"><p>
 353           <span class="inlinemediaobject"><img src="../../../equations/heuman_lambda.svg"></span>
 354
 355         </p></blockquote></div>
 356 <p>
 357         Both of these functions are easily implemented in terms of Carlson's integrals,
 358         and are provided in this library as <a class="link" href="jacobi_zeta.html" title="Jacobi Zeta Function">jacobi_zeta</a>
 359         and <a class="link" href="heuman_lambda.html" title="Heuman Lambda Function">heuman_lambda</a>.
 360       </p>
 361 <h5>
 362 <a name="math_toolkit.ellint.ellint_intro.h6"></a>
 363         <span class="phrase"><a name="math_toolkit.ellint.ellint_intro.numerical_algorithms"></a></span><a class="link" href="ellint_intro.html#math_toolkit.ellint.ellint_intro.numerical_algorithms">Numerical
 364         Algorithms</a>
 365       </h5>
 366 <p>
 367         The conventional methods for computing elliptic integrals are Gauss and Landen
 368         transformations, which converge quadratically and work well for elliptic
 369         integrals of the first and second kinds. Unfortunately they suffer from loss
 370         of significant digits for the third kind.
 371       </p>
 372 <p>
 373         Carlson's algorithm [<a class="link" href="ellint_intro.html#ellint_ref_carlson79">Carlson79</a>]
 374         [<a class="link" href="ellint_intro.html#ellint_ref_carlson78">Carlson78</a>], by contrast, provides
 375         a unified method for all three kinds of elliptic integrals with satisfactory
 376         precisions.
 377       </p>
 378 <h5>
 379 <a name="math_toolkit.ellint.ellint_intro.h7"></a>
 380         <span class="phrase"><a name="math_toolkit.ellint.ellint_intro.references"></a></span><a class="link" href="ellint_intro.html#math_toolkit.ellint.ellint_intro.references">References</a>
 381       </h5>
 382 <p>
 383         Special mention goes to:
 384       </p>
 385 <div class="blockquote"><blockquote class="blockquote"><p>
 386           A. M. Legendre, <span class="emphasis"><em>Trait&#233; des Fonctions Elliptiques et des Integrales
 387           Euleriennes</em></span>, Vol. 1. Paris (1825).
 388         </p></blockquote></div>
 389 <p>
 390         However the main references are:
 391       </p>
 392 <div class="orderedlist"><ol class="orderedlist" type="1">
 393 <li class="listitem">
 394             <a name="ellint_ref_AS"></a>M. Abramowitz and I. A. Stegun (Eds.) (1964)
 395             Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical
 396             Tables, National Bureau of Standards Applied Mathematics Series, U.S.
 397             Government Printing Office, Washington, D.C.
 398           </li>
 399 <li class="listitem">
 400             <a href="https://dlmf.nist.gov/19" target="_top">NIST Digital Library of Mathematical
 401             Functions, Elliptic Integrals, B. C. Carlson</a>
 402           </li>
 403 <li class="listitem">
 404             <a name="ellint_ref_carlson79"></a>B.C. Carlson, <span class="emphasis"><em>Computing
 405             elliptic integrals by duplication</em></span>, Numerische Mathematik,
 406             vol 33, 1 (1979).
 407           </li>
 408 <li class="listitem">
 409             <a name="ellint_ref_carlson77"></a>B.C. Carlson, <span class="emphasis"><em>Elliptic Integrals
 410             of the First Kind</em></span>, SIAM Journal on Mathematical Analysis,
 411             vol 8, 231 (1977).
 412           </li>
 413 <li class="listitem">
 414             <a name="ellint_ref_carlson78"></a>B.C. Carlson, <span class="emphasis"><em>Short Proofs
 415             of Three Theorems on Elliptic Integrals</em></span>, SIAM Journal on Mathematical
 416             Analysis, vol 9, 524 (1978).
 417           </li>
 418 <li class="listitem">
 419             <a name="ellint_ref_carlson81"></a>B.C. Carlson and E.M. Notis, <span class="emphasis"><em>ALGORITHM
 420             577: Algorithms for Incomplete Elliptic Integrals</em></span>, ACM Transactions
 421             on Mathematmal Software, vol 7, 398 (1981).
 422           </li>
 423 <li class="listitem">
 424             B. C. Carlson, <span class="emphasis"><em>On computing elliptic integrals and functions</em></span>.
 425             J. Math. and Phys., 44 (1965), pp. 36-51.
 426           </li>
 427 <li class="listitem">
 428             B. C. Carlson, <span class="emphasis"><em>A table of elliptic integrals of the second
 429             kind</em></span>. Math. Comp., 49 (1987), pp. 595-606. (Supplement, ibid.,
 430             pp. S13-S17.)
 431           </li>
 432 <li class="listitem">
 433             B. C. Carlson, <span class="emphasis"><em>A table of elliptic integrals of the third kind</em></span>.
 434             Math. Comp., 51 (1988), pp. 267-280. (Supplement, ibid., pp. S1-S5.)
 435           </li>
 436 <li class="listitem">
 437             B. C. Carlson, <span class="emphasis"><em>A table of elliptic integrals: cubic cases</em></span>.
 438             Math. Comp., 53 (1989), pp. 327-333.
 439           </li>
 440 <li class="listitem">
 441             B. C. Carlson, <span class="emphasis"><em>A table of elliptic integrals: one quadratic
 442             factor</em></span>. Math. Comp., 56 (1991), pp. 267-280.
 443           </li>
 444 <li class="listitem">
 445             B. C. Carlson, <span class="emphasis"><em>A table of elliptic integrals: two quadratic
 446             factors</em></span>. Math. Comp., 59 (1992), pp. 165-180.
 447           </li>
 448 <li class="listitem">
 449             B. C. Carlson, <span class="emphasis"><em><a href="http://arxiv.org/abs/math.CA/9409227" target="_top">Numerical
 450             computation of real or complex elliptic integrals</a></em></span>.
 451             Numerical Algorithms, Volume 10, Number 1 / March, 1995, p13-26.
 452           </li>
 453 <li class="listitem">
 454             B. C. Carlson and John L. Gustafson, <span class="emphasis"><em><a href="http://arxiv.org/abs/math.CA/9310223" target="_top">Asymptotic
 455             Approximations for Symmetric Elliptic Integrals</a></em></span>, SIAM
 456             Journal on Mathematical Analysis, Volume 25, Issue 2 (March 1994), 288-303.
 457           </li>
 458 </ol></div>
 459 <p>
 460         The following references, while not directly relevent to our implementation,
 461         may also be of interest:
 462       </p>
 463 <div class="orderedlist"><ol class="orderedlist" type="1">
 464 <li class="listitem">
 465             R. Burlisch, <span class="emphasis"><em>Numerical Compuation of Elliptic Integrals and
 466             Elliptic Functions.</em></span> Numerical Mathematik 7, 78-90.
 467           </li>
 468 <li class="listitem">
 469             R. Burlisch, <span class="emphasis"><em>An extension of the Bartky Transformation to Incomplete
 470             Elliptic Integrals of the Third Kind</em></span>. Numerical Mathematik
 471             13, 266-284.
 472           </li>
 473 <li class="listitem">
 474             R. Burlisch, <span class="emphasis"><em>Numerical Compuation of Elliptic Integrals and
 475             Elliptic Functions. III</em></span>. Numerical Mathematik 13, 305-315.
 476           </li>
 477 <li class="listitem">
 478             T. Fukushima and H. Ishizaki, <span class="emphasis"><em><a href="http://adsabs.harvard.edu/abs/1994CeMDA..59..237F" target="_top">Numerical
 479             Computation of Incomplete Elliptic Integrals of a General Form.</a></em></span>
 480             Celestial Mechanics and Dynamical Astronomy, Volume 59, Number 3 / July,
 481             1994, 237-251.
 482           </li>
 483 </ol></div>
 484 </div>
 485 <table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr>
 486 <td align="left"></td>
 487 <td align="right"><div class="copyright-footer">Copyright &#169; 2006-2019 Nikhar
 488       Agrawal, Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos,
 489       Hubert Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Matthew Pulver, Johan
 490       R&#229;de, Gautam Sewani, Benjamin Sobotta, Nicholas Thompson, Thijs van den Berg,
 491       Daryle Walker and Xiaogang Zhang<p>
 492         Distributed under the Boost Software License, Version 1.0. (See accompanying
 493         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>)
 494       </p>
 495 </div></td>
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