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Backgrounders"> 10<link rel="next" href="special_tut.html" title="Tutorial: How to Write a New Special Function"> 11</head> 12<body bgcolor="white" text="black" link="#0000FF" vlink="#840084" alink="#0000FF"> 13<table cellpadding="2" width="100%"><tr> 14<td valign="top"><img alt="Boost C++ Libraries" width="277" height="86" src="../../../../../boost.png"></td> 15<td align="center"><a href="../../../../../index.html">Home</a></td> 16<td align="center"><a href="../../../../../libs/libraries.htm">Libraries</a></td> 17<td align="center"><a href="http://www.boost.org/users/people.html">People</a></td> 18<td align="center"><a href="http://www.boost.org/users/faq.html">FAQ</a></td> 19<td align="center"><a href="../../../../../more/index.htm">More</a></td> 20</tr></table> 21<hr> 22<div class="spirit-nav"> 23<a accesskey="p" href="../backgrounders.html"><img src="../../../../../doc/src/images/prev.png" alt="Prev"></a><a accesskey="u" href="../backgrounders.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="special_tut.html"><img src="../../../../../doc/src/images/next.png" alt="Next"></a> 24</div> 25<div class="section"> 26<div class="titlepage"><div><div><h2 class="title" style="clear: both"> 27<a name="math_toolkit.sf_implementation"></a><a class="link" href="sf_implementation.html" title="Additional Implementation Notes">Additional Implementation 28 Notes</a> 29</h2></div></div></div> 30<p> 31 The majority of the implementation notes are included with the documentation 32 of each function or distribution. The notes here are of a more general nature, 33 and reflect more the general implementation philosophy used. 34 </p> 35<h5> 36<a name="math_toolkit.sf_implementation.h0"></a> 37 <span class="phrase"><a name="math_toolkit.sf_implementation.implementation_philosophy"></a></span><a class="link" href="sf_implementation.html#math_toolkit.sf_implementation.implementation_philosophy">Implementation 38 philosophy</a> 39 </h5> 40<p> 41 "First be right, then be fast." 42 </p> 43<p> 44 There will always be potential compromises to be made between speed and accuracy. 45 It may be possible to find faster methods, particularly for certain limited 46 ranges of arguments, but for most applications of math functions and distributions, 47 we judge that speed is rarely as important as accuracy. 48 </p> 49<p> 50 So our priority is accuracy. 51 </p> 52<p> 53 To permit evaluation of accuracy of the special functions, production of extremely 54 accurate tables of test values has received considerable effort. 55 </p> 56<p> 57 (It also required much CPU effort - there was some danger of molten plastic 58 dripping from the bottom of JM's laptop, so instead, PAB's Dual-core desktop 59 was kept 50% busy for <span class="bold"><strong>days</strong></span> calculating some 60 tables of test values!) 61 </p> 62<p> 63 For a specific RealType, say <code class="computeroutput"><span class="keyword">float</span></code> 64 or <code class="computeroutput"><span class="keyword">double</span></code>, it may be possible 65 to find approximations for some functions that are simpler and thus faster, 66 but less accurate (perhaps because there are no refining iterations, for example, 67 when calculating inverse functions). 68 </p> 69<p> 70 If these prove accurate enough to be "fit for his purpose", then 71 a user may substitute his custom specialization. 72 </p> 73<p> 74 For example, there are approximations dating back from times when computation 75 was a <span class="bold"><strong>lot</strong></span> more expensive: 76 </p> 77<p> 78 H Goldberg and H Levine, Approximate formulas for percentage points and normalisation 79 of t and chi squared, Ann. Math. Stat., 17(4), 216 - 225 (Dec 1946). 80 </p> 81<p> 82 A H Carter, Approximations to percentage points of the z-distribution, Biometrika 83 34(2), 352 - 358 (Dec 1947). 84 </p> 85<p> 86 These could still provide sufficient accuracy for some speed-critical applications. 87 </p> 88<h5> 89<a name="math_toolkit.sf_implementation.h1"></a> 90 <span class="phrase"><a name="math_toolkit.sf_implementation.accuracy_and_representation_of_t"></a></span><a class="link" href="sf_implementation.html#math_toolkit.sf_implementation.accuracy_and_representation_of_t">Accuracy 91 and Representation of Test Values</a> 92 </h5> 93<p> 94 In order to be accurate enough for as many as possible real types, constant 95 values are given to 50 decimal digits if available (though many sources proved 96 only accurate near to 64-bit double precision). Values are specified as long 97 double types by appending L, unless they are exactly representable, for example 98 integers, or binary fractions like 0.125. This avoids the risk of loss of accuracy 99 converting from double, the default type. Values are used after <code class="computeroutput"><span class="keyword">static_cast</span><span class="special"><</span><span class="identifier">RealType</span><span class="special">>(</span><span class="number">1.2345L</span><span class="special">)</span></code> to provide 100 the appropriate RealType for spot tests. 101 </p> 102<p> 103 Functions that return constants values, like kurtosis for example, are written 104 as 105 </p> 106<p> 107 <code class="computeroutput"><span class="keyword">static_cast</span><span class="special"><</span><span class="identifier">RealType</span><span class="special">>(-</span><span class="number">3</span><span class="special">)</span> <span class="special">/</span> 108 <span class="number">5</span><span class="special">;</span></code> 109 </p> 110<p> 111 to provide the most accurate value that the compiler can compute for the real 112 type. (The denominator is an integer and so will be promoted exactly). 113 </p> 114<p> 115 So tests for one third, <span class="bold"><strong>not</strong></span> exactly representable 116 with radix two floating-point, (should) use, for example: 117 </p> 118<p> 119 <code class="computeroutput"><span class="keyword">static_cast</span><span class="special"><</span><span class="identifier">RealType</span><span class="special">>(</span><span class="number">1</span><span class="special">)</span> <span class="special">/</span> 120 <span class="number">3</span><span class="special">;</span></code> 121 </p> 122<p> 123 If a function is very sensitive to changes in input, specifying an inexact 124 value as input (such as 0.1) can throw the result off by a noticeable amount: 125 0.1f is "wrong" by ~1e-7 for example (because 0.1 has no exact binary 126 representation). That is why exact binary values - halves, quarters, and eighths 127 etc - are used in test code along with the occasional fraction <code class="computeroutput"><span class="identifier">a</span><span class="special">/</span><span class="identifier">b</span></code> 128 with <code class="computeroutput"><span class="identifier">b</span></code> a power of two (in order 129 to ensure that the result is an exactly representable binary value). 130 </p> 131<h5> 132<a name="math_toolkit.sf_implementation.h2"></a> 133 <span class="phrase"><a name="math_toolkit.sf_implementation.tolerance_of_tests"></a></span><a class="link" href="sf_implementation.html#math_toolkit.sf_implementation.tolerance_of_tests">Tolerance 134 of Tests</a> 135 </h5> 136<p> 137 The tolerances need to be set to the maximum of: 138 </p> 139<div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: disc; "> 140<li class="listitem"> 141 Some epsilon value. 142 </li> 143<li class="listitem"> 144 The accuracy of the data (often only near 64-bit double). 145 </li> 146</ul></div> 147<p> 148 Otherwise when long double has more digits than the test data, then no amount 149 of tweaking an epsilon based tolerance will work. 150 </p> 151<p> 152 A common problem is when tolerances that are suitable for implementations like 153 Microsoft VS.NET where double and long double are the same size: tests fail 154 on other systems where long double is more accurate than double. Check first 155 that the suffix L is present, and then that the tolerance is big enough. 156 </p> 157<h5> 158<a name="math_toolkit.sf_implementation.h3"></a> 159 <span class="phrase"><a name="math_toolkit.sf_implementation.handling_unsuitable_arguments"></a></span><a class="link" href="sf_implementation.html#math_toolkit.sf_implementation.handling_unsuitable_arguments">Handling 160 Unsuitable Arguments</a> 161 </h5> 162<p> 163 In <a href="http://www.open-std.org/jtc1/sc22/wg21/docs/papers/2004/n1665.pdf" target="_top">Errors 164 in Mathematical Special Functions</a>, J. Marraffino & M. Paterno it 165 is proposed that signalling a domain error is mandatory when the argument would 166 give an mathematically undefined result. 167 </p> 168<div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: disc; "><li class="listitem"> 169 Guideline 1 170 </li></ul></div> 171<div class="blockquote"><blockquote class="blockquote"><p> 172 A mathematical function is said to be defined at a point a = (a1, a2, . . 173 .) if the limits as x = (x1, x2, . . .) 'approaches a from all directions 174 agree'. The defined value may be any number, or +infinity, or -infinity. 175 </p></blockquote></div> 176<p> 177 Put crudely, if the function goes to + infinity and then emerges 'round-the-back' 178 with - infinity, it is NOT defined. 179 </p> 180<div class="blockquote"><blockquote class="blockquote"><p> 181 The library function which approximates a mathematical function shall signal 182 a domain error whenever evaluated with argument values for which the mathematical 183 function is undefined. 184 </p></blockquote></div> 185<div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: disc; "><li class="listitem"> 186 Guideline 2 187 </li></ul></div> 188<div class="blockquote"><blockquote class="blockquote"><p> 189 The library function which approximates a mathematical function shall signal 190 a domain error whenever evaluated with argument values for which the mathematical 191 function obtains a non-real value. 192 </p></blockquote></div> 193<p> 194 This implementation is believed to follow these proposals and to assist compatibility 195 with <span class="emphasis"><em>ISO/IEC 9899:1999 Programming languages - C</em></span> and with 196 the <a href="http://www.open-std.org/jtc1/sc22/wg21/docs/papers/2005/n1836.pdf" target="_top">Draft 197 Technical Report on C++ Library Extensions, 2005-06-24, section 5.2.1, paragraph 198 5</a>. <a class="link" href="error_handling.html" title="Error Handling">See also domain_error</a>. 199 </p> 200<p> 201 See <a class="link" href="pol_ref.html" title="Policy Reference">policy reference</a> for details 202 of the error handling policies that should allow a user to comply with any 203 of these recommendations, as well as other behaviour. 204 </p> 205<p> 206 See <a class="link" href="error_handling.html" title="Error Handling">error handling</a> for a 207 detailed explanation of the mechanism, and <a class="link" href="stat_tut/weg/error_eg.html" title="Error Handling Example">error_handling 208 example</a> and <a href="../../../example/error_handling_example.cpp" target="_top">error_handling_example.cpp</a> 209 </p> 210<div class="caution"><table border="0" summary="Caution"> 211<tr> 212<td rowspan="2" align="center" valign="top" width="25"><img alt="[Caution]" src="../../../../../doc/src/images/caution.png"></td> 213<th align="left">Caution</th> 214</tr> 215<tr><td align="left" valign="top"><p> 216 If you enable throw but do NOT have try & catch block, then the program 217 will terminate with an uncaught exception and probably abort. Therefore to 218 get the benefit of helpful error messages, enabling <span class="bold"><strong>all</strong></span> 219 exceptions <span class="bold"><strong>and</strong></span> using try&catch is recommended 220 for all applications. However, for simplicity, this is not done for most 221 examples. 222 </p></td></tr> 223</table></div> 224<h5> 225<a name="math_toolkit.sf_implementation.h4"></a> 226 <span class="phrase"><a name="math_toolkit.sf_implementation.handling_of_functions_that_are_n"></a></span><a class="link" href="sf_implementation.html#math_toolkit.sf_implementation.handling_of_functions_that_are_n">Handling 227 of Functions that are Not Mathematically defined</a> 228 </h5> 229<p> 230 Functions that are not mathematically defined, like the Cauchy mean, fail to 231 compile by default. A <a class="link" href="pol_ref/assert_undefined.html" title="Mathematically Undefined Function Policies">policy</a> 232 allows control of this. 233 </p> 234<p> 235 If the policy is to permit undefined functions, then calling them throws a 236 domain error, by default. But the error policy can be set to not throw, and 237 to return NaN instead. For example, 238 </p> 239<p> 240 <code class="computeroutput"><span class="preprocessor">#define</span> <span class="identifier">BOOST_MATH_DOMAIN_ERROR_POLICY</span> 241 <span class="identifier">ignore_error</span></code> 242 </p> 243<p> 244 appears before the first Boost include, then if the un-implemented function 245 is called, mean(cauchy<>()) will return std::numeric_limits<T>::quiet_NaN(). 246 </p> 247<div class="warning"><table border="0" summary="Warning"> 248<tr> 249<td rowspan="2" align="center" valign="top" width="25"><img alt="[Warning]" src="../../../../../doc/src/images/warning.png"></td> 250<th align="left">Warning</th> 251</tr> 252<tr><td align="left" valign="top"><p> 253 If <code class="computeroutput"><span class="identifier">std</span><span class="special">::</span><span class="identifier">numeric_limits</span><span class="special"><</span><span class="identifier">T</span><span class="special">>::</span><span class="identifier">has_quiet_NaN</span></code> is false (for example, if 254 T is a User-defined type without NaN support), then an exception will always 255 be thrown when a domain error occurs. Catching exceptions is therefore strongly 256 recommended. 257 </p></td></tr> 258</table></div> 259<h5> 260<a name="math_toolkit.sf_implementation.h5"></a> 261 <span class="phrase"><a name="math_toolkit.sf_implementation.median_of_distributions"></a></span><a class="link" href="sf_implementation.html#math_toolkit.sf_implementation.median_of_distributions">Median of 262 distributions</a> 263 </h5> 264<p> 265 There are many distributions for which we have been unable to find an analytic 266 formula, and this has deterred us from implementing <a href="http://en.wikipedia.org/wiki/Median" target="_top">median 267 functions</a>, the mid-point in a list of values. 268 </p> 269<p> 270 However a useful numerical approximation for distribution <code class="computeroutput"><span class="identifier">dist</span></code> 271 is available as usual as an accessor non-member function median using <code class="computeroutput"><span class="identifier">median</span><span class="special">(</span><span class="identifier">dist</span><span class="special">)</span></code>, that may be evaluated (in the absence of 272 an analytic formula) by calling 273 </p> 274<p> 275 <code class="computeroutput"><span class="identifier">quantile</span><span class="special">(</span><span class="identifier">dist</span><span class="special">,</span> <span class="number">0.5</span><span class="special">)</span></code> (this is the <span class="emphasis"><em>mathematical</em></span> 276 definition of course). 277 </p> 278<p> 279 <a href="http://www.amstat.org/publications/jse/v13n2/vonhippel.html" target="_top">Mean, 280 Median, and Skew, Paul T von Hippel</a> 281 </p> 282<p> 283 <a href="http://documents.wolfram.co.jp/teachersedition/MathematicaBook/24.5.html" target="_top">Descriptive 284 Statistics,</a> 285 </p> 286<p> 287 <a href="http://documents.wolfram.co.jp/v5/Add-onsLinks/StandardPackages/Statistics/DescriptiveStatistics.html" target="_top">and 288 </a> 289 </p> 290<p> 291 <a href="http://documents.wolfram.com/v5/TheMathematicaBook/AdvancedMathematicsInMathematica/NumericalOperationsOnData/3.8.1.html" target="_top">Mathematica 292 Basic Statistics.</a> give more detail, in particular for discrete distributions. 293 </p> 294<h5> 295<a name="math_toolkit.sf_implementation.h6"></a> 296 <span class="phrase"><a name="math_toolkit.sf_implementation.handling_of_floating_point_infin"></a></span><a class="link" href="sf_implementation.html#math_toolkit.sf_implementation.handling_of_floating_point_infin">Handling 297 of Floating-Point Infinity</a> 298 </h5> 299<p> 300 Some functions and distributions are well defined with + or - infinity as argument(s), 301 but after some experiments with handling infinite arguments as special cases, 302 we concluded that it was generally more useful to forbid this, and instead 303 to return the result of <a class="link" href="error_handling.html#math_toolkit.error_handling.domain_error">domain_error</a>. 304 </p> 305<p> 306 Handling infinity as special cases is additionally complicated because, unlike 307 built-in types on most - but not all - platforms, not all User-Defined Types 308 are specialized to provide <code class="computeroutput"><span class="identifier">std</span><span class="special">::</span><span class="identifier">numeric_limits</span><span class="special"><</span><span class="identifier">RealType</span><span class="special">>::</span><span class="identifier">infinity</span><span class="special">()</span></code> and would return zero rather than any representation 309 of infinity. 310 </p> 311<p> 312 The rationale is that non-finiteness may happen because of error or overflow 313 in the users code, and it will be more helpful for this to be diagnosed promptly 314 rather than just continuing. The code also became much more complicated, more 315 error-prone, much more work to test, and much less readable. 316 </p> 317<p> 318 However in a few cases, for example normal, where we felt it obvious, we have 319 permitted argument(s) to be infinity, provided infinity is implemented for 320 the <code class="computeroutput"><span class="identifier">RealType</span></code> on that implementation, 321 and it is supported and tested by the distribution. 322 </p> 323<p> 324 The range for these distributions is set to infinity if supported by the platform, 325 (by testing <code class="computeroutput"><span class="identifier">std</span><span class="special">::</span><span class="identifier">numeric_limits</span><span class="special"><</span><span class="identifier">RealType</span><span class="special">>::</span><span class="identifier">has_infinity</span></code>) else the maximum value provided 326 for the <code class="computeroutput"><span class="identifier">RealType</span></code> by Boost.Math. 327 </p> 328<p> 329 Testing for has_infinity is obviously important for arbitrary precision types 330 where infinity makes much less sense than for IEEE754 floating-point. 331 </p> 332<p> 333 So far we have not set <code class="computeroutput"><span class="identifier">support</span><span class="special">()</span></code> function (only range) on the grounds that 334 the PDF is uninteresting/zero for infinities. 335 </p> 336<p> 337 Users who require special handling of infinity (or other specific value) can, 338 of course, always intercept this before calling a distribution or function 339 and return their own choice of value, or other behavior. This will often be 340 simpler than trying to handle the aftermath of the error policy. 341 </p> 342<p> 343 Overflow, underflow, denorm can be handled using <a class="link" href="pol_ref/error_handling_policies.html" title="Error Handling Policies">error 344 handling policies</a>. 345 </p> 346<p> 347 We have also tried to catch boundary cases where the mathematical specification 348 would result in divide by zero or overflow and signalling these similarly. 349 What happens at (and near), poles can be controlled through <a class="link" href="pol_ref/error_handling_policies.html" title="Error Handling Policies">error 350 handling policies</a>. 351 </p> 352<h5> 353<a name="math_toolkit.sf_implementation.h7"></a> 354 <span class="phrase"><a name="math_toolkit.sf_implementation.scale_shape_and_location"></a></span><a class="link" href="sf_implementation.html#math_toolkit.sf_implementation.scale_shape_and_location">Scale, Shape 355 and Location</a> 356 </h5> 357<p> 358 We considered adding location and scale to the list of functions, for example: 359 </p> 360<pre class="programlisting"><span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">RealType</span><span class="special">></span> 361<span class="keyword">inline</span> <span class="identifier">RealType</span> <span class="identifier">scale</span><span class="special">(</span><span class="keyword">const</span> <span class="identifier">triangular_distribution</span><span class="special"><</span><span class="identifier">RealType</span><span class="special">>&</span> <span class="identifier">dist</span><span class="special">)</span> 362<span class="special">{</span> 363 <span class="identifier">RealType</span> <span class="identifier">lower</span> <span class="special">=</span> <span class="identifier">dist</span><span class="special">.</span><span class="identifier">lower</span><span class="special">();</span> 364 <span class="identifier">RealType</span> <span class="identifier">mode</span> <span class="special">=</span> <span class="identifier">dist</span><span class="special">.</span><span class="identifier">mode</span><span class="special">();</span> 365 <span class="identifier">RealType</span> <span class="identifier">upper</span> <span class="special">=</span> <span class="identifier">dist</span><span class="special">.</span><span class="identifier">upper</span><span class="special">();</span> 366 <span class="identifier">RealType</span> <span class="identifier">result</span><span class="special">;</span> <span class="comment">// of checks.</span> 367 <span class="keyword">if</span><span class="special">(</span><span class="keyword">false</span> <span class="special">==</span> <span class="identifier">detail</span><span class="special">::</span><span class="identifier">check_triangular</span><span class="special">(</span><span class="identifier">BOOST_CURRENT_FUNCTION</span><span class="special">,</span> <span class="identifier">lower</span><span class="special">,</span> <span class="identifier">mode</span><span class="special">,</span> <span class="identifier">upper</span><span class="special">,</span> <span class="special">&</span><span class="identifier">result</span><span class="special">))</span> 368 <span class="special">{</span> 369 <span class="keyword">return</span> <span class="identifier">result</span><span class="special">;</span> 370 <span class="special">}</span> 371 <span class="keyword">return</span> <span class="special">(</span><span class="identifier">upper</span> <span class="special">-</span> <span class="identifier">lower</span><span class="special">);</span> 372<span class="special">}</span> 373</pre> 374<p> 375 but found that these concepts are not defined (or their definition too contentious) 376 for too many distributions to be generally applicable. Because they are non-member 377 functions, they can be added if required. 378 </p> 379<h5> 380<a name="math_toolkit.sf_implementation.h8"></a> 381 <span class="phrase"><a name="math_toolkit.sf_implementation.notes_on_implementation_of_speci"></a></span><a class="link" href="sf_implementation.html#math_toolkit.sf_implementation.notes_on_implementation_of_speci">Notes 382 on Implementation of Specific Functions & Distributions</a> 383 </h5> 384<div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: disc; "><li class="listitem"> 385 Default parameters for the Triangular Distribution. We are uncertain about 386 the best default parameters. Some sources suggest that the Standard Triangular 387 Distribution has lower = 0, mode = half and upper = 1. However as a approximation 388 for the normal distribution, the most common usage, lower = -1, mode = 389 0 and upper = 1 would be more suitable. 390 </li></ul></div> 391<h5> 392<a name="math_toolkit.sf_implementation.h9"></a> 393 <span class="phrase"><a name="math_toolkit.sf_implementation.rational_approximations_used"></a></span><a class="link" href="sf_implementation.html#math_toolkit.sf_implementation.rational_approximations_used">Rational 394 Approximations Used</a> 395 </h5> 396<p> 397 Some of the special functions in this library are implemented via rational 398 approximations. These are either taken from the literature, or devised by John 399 Maddock using <a class="link" href="internals/minimax.html" title="Minimax Approximations and the Remez Algorithm">our Remez code</a>. 400 </p> 401<p> 402 Rational rather than Polynomial approximations are used to ensure accuracy: 403 polynomial approximations are often wonderful up to a certain level of accuracy, 404 but then quite often fail to provide much greater accuracy no matter how many 405 more terms are added. 406 </p> 407<p> 408 Our own approximations were devised either for added accuracy (to support 128-bit 409 long doubles for example), or because literature methods were unavailable or 410 under non-BSL compatible license. Our Remez code is known to produce good agreement 411 with literature results in fairly simple "toy" cases. All approximations 412 were checked for convergence and to ensure that they were not ill-conditioned 413 (the coefficients can give a theoretically good solution, but the resulting 414 rational function may be un-computable at fixed precision). 415 </p> 416<p> 417 Recomputing using different Remez implementations may well produce differing 418 coefficients: the problem is well known to be ill conditioned in general, and 419 our Remez implementation often found a broad and ill-defined minima for many 420 of these approximations (of course for simple "toy" examples like 421 approximating <code class="computeroutput"><span class="identifier">exp</span></code> the minima 422 is well defined, and the coefficients should agree no matter whose Remez implementation 423 is used). This should not in general effect the validity of the approximations: 424 there's good literature supporting the idea that coefficients can be "in 425 error" without necessarily adversely effecting the result. Note that "in 426 error" has a special meaning in this context, see <a href="http://front.math.ucdavis.edu/0101.5042" target="_top">"Approximate 427 construction of rational approximations and the effect of error autocorrection.", 428 Grigori Litvinov, eprint arXiv:math/0101042</a>. Therefore the coefficients 429 still need to be accurately calculated, even if they can be in error compared 430 to the "true" minimax solution. 431 </p> 432<h5> 433<a name="math_toolkit.sf_implementation.h10"></a> 434 <span class="phrase"><a name="math_toolkit.sf_implementation.representation_of_mathematical_c"></a></span><a class="link" href="sf_implementation.html#math_toolkit.sf_implementation.representation_of_mathematical_c">Representation 435 of Mathematical Constants</a> 436 </h5> 437<p> 438 A macro BOOST_DEFINE_MATH_CONSTANT in constants.hpp is used to provide high 439 accuracy constants to mathematical functions and distributions, since it is 440 important to provide values uniformly for both built-in float, double and long 441 double types, and for User Defined types in <a href="../../../../../libs/multiprecision/doc/html/index.html" target="_top">Boost.Multiprecision</a> 442 like <a href="../../../../../libs/multiprecision/doc/html/boost_multiprecision/tut/floats/cpp_dec_float.html" target="_top">cpp_dec_float</a>. 443 and others like NTL::quad_float and NTL::RR. 444 </p> 445<p> 446 To permit calculations in this Math ToolKit and its tests, (and elsewhere) 447 at about 100 decimal digits with NTL::RR type, it is obviously necessary to 448 define constants to this accuracy. 449 </p> 450<p> 451 However, some compilers do not accept decimal digits strings as long as this. 452 So the constant is split into two parts, with the 1st containing at least long 453 double precision, and the 2nd zero if not needed or known. The 3rd part permits 454 an exponent to be provided if necessary (use zero if none) - the other two 455 parameters may only contain decimal digits (and sign and decimal point), and 456 may NOT include an exponent like 1.234E99 (nor a trailing F or L). The second 457 digit string is only used if T is a User-Defined Type, when the constant is 458 converted to a long string literal and lexical_casted to type T. (This is necessary 459 because you can't use a numeric constant since even a long double might not 460 have enough digits). 461 </p> 462<p> 463 For example, pi is defined: 464 </p> 465<pre class="programlisting"><span class="identifier">BOOST_DEFINE_MATH_CONSTANT</span><span class="special">(</span><span class="identifier">pi</span><span class="special">,</span> 466 <span class="number">3.141592653589793238462643383279502884197169399375105820974944</span><span class="special">,</span> 467 <span class="number">5923078164062862089986280348253421170679821480865132823066470938446095505</span><span class="special">,</span> 468 <span class="number">0</span><span class="special">)</span> 469</pre> 470<p> 471 And used thus: 472 </p> 473<pre class="programlisting"><span class="keyword">using</span> <span class="keyword">namespace</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">::</span><span class="identifier">constants</span><span class="special">;</span> 474 475<span class="keyword">double</span> <span class="identifier">diameter</span> <span class="special">=</span> <span class="number">1.</span><span class="special">;</span> 476<span class="keyword">double</span> <span class="identifier">radius</span> <span class="special">=</span> <span class="identifier">diameter</span> <span class="special">*</span> <span class="identifier">pi</span><span class="special"><</span><span class="keyword">double</span><span class="special">>();</span> 477 478<span class="keyword">or</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">::</span><span class="identifier">constants</span><span class="special">::</span><span class="identifier">pi</span><span class="special"><</span><span class="identifier">NTL</span><span class="special">::</span><span class="identifier">RR</span><span class="special">>()</span> 479</pre> 480<p> 481 Note that it is necessary (if inconvenient) to specify the type explicitly. 482 </p> 483<p> 484 So you cannot write 485 </p> 486<pre class="programlisting"><span class="keyword">double</span> <span class="identifier">p</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">constants</span><span class="special">::</span><span class="identifier">pi</span><span class="special"><>();</span> <span class="comment">// could not deduce template argument for 'T'</span> 487</pre> 488<p> 489 Neither can you write: 490 </p> 491<pre class="programlisting"><span class="keyword">double</span> <span class="identifier">p</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">constants</span><span class="special">::</span><span class="identifier">pi</span><span class="special">;</span> <span class="comment">// Context does not allow for disambiguation of overloaded function</span> 492<span class="keyword">double</span> <span class="identifier">p</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">constants</span><span class="special">::</span><span class="identifier">pi</span><span class="special">();</span> <span class="comment">// Context does not allow for disambiguation of overloaded function</span> 493</pre> 494<h5> 495<a name="math_toolkit.sf_implementation.h11"></a> 496 <span class="phrase"><a name="math_toolkit.sf_implementation.thread_safety"></a></span><a class="link" href="sf_implementation.html#math_toolkit.sf_implementation.thread_safety">Thread 497 safety</a> 498 </h5> 499<p> 500 Reporting of error by setting <code class="computeroutput"><span class="identifier">errno</span></code> 501 should be thread-safe already (otherwise none of the std lib math functions 502 would be thread safe?). If you turn on reporting of errors via exceptions, 503 <code class="computeroutput"><span class="identifier">errno</span></code> gets left unused anyway. 504 </p> 505<p> 506 For normal C++ usage, the Boost.Math <code class="computeroutput"><span class="keyword">static</span> 507 <span class="keyword">const</span></code> constants are now thread-safe 508 so for built-in real-number types: <code class="computeroutput"><span class="keyword">float</span></code>, 509 <code class="computeroutput"><span class="keyword">double</span></code> and <code class="computeroutput"><span class="keyword">long</span> 510 <span class="keyword">double</span></code> are all thread safe. 511 </p> 512<p> 513 For User_defined types, for example, <a href="../../../../../libs/multiprecision/doc/html/boost_multiprecision/tut/floats/cpp_dec_float.html" target="_top">cpp_dec_float</a>, 514 the Boost.Math should also be thread-safe, (thought we are unsure how to rigorously 515 prove this). 516 </p> 517<p> 518 (Thread safety has received attention in the C++11 Standard revision, so hopefully 519 all compilers will do the right thing here at some point.) 520 </p> 521<h5> 522<a name="math_toolkit.sf_implementation.h12"></a> 523 <span class="phrase"><a name="math_toolkit.sf_implementation.sources_of_test_data"></a></span><a class="link" href="sf_implementation.html#math_toolkit.sf_implementation.sources_of_test_data">Sources 524 of Test Data</a> 525 </h5> 526<p> 527 We found a large number of sources of test data. We have assumed that these 528 are <span class="emphasis"><em>"known good"</em></span> if they agree with the results 529 from our test and only consulted other sources for their <span class="emphasis"><em>'vote'</em></span> 530 in the case of serious disagreement. The accuracy, actual and claimed, vary 531 very widely. Only <a href="http://functions.wolfram.com/" target="_top">Wolfram Mathematica 532 functions</a> provided a higher accuracy than C++ double (64-bit floating-point) 533 and was regarded as the most-trusted source by far. The <a href="http://www.r-project.org/" target="_top">The 534 R Project for Statistical Computing</a> provided the widest range of distributions, 535 but the usual Intel X86 distribution uses 64-but doubles, so our use was limited 536 to the 15 to 17 decimal digit accuracy. 537 </p> 538<p> 539 A useful index of sources is: <a href="http://www.sal.hut.fi/Teaching/Resources/ProbStat/table.html" target="_top">Web-oriented 540 Teaching Resources in Probability and Statistics</a> 541 </p> 542<p> 543 <a href="http://espse.ed.psu.edu/edpsych/faculty/rhale/hale/507Mat/statlets/free/pdist.htm" target="_top">Statlet</a>: 544 Is a Javascript application that calculates and plots probability distributions, 545 and provides the most complete range of distributions: 546 </p> 547<div class="blockquote"><blockquote class="blockquote"><p> 548 Bernoulli, Binomial, discrete uniform, geometric, hypergeometric, negative 549 binomial, Poisson, beta, Cauchy-Lorentz, chi-squared, Erlang, exponential, 550 extreme value, Fisher, gamma, Laplace, logistic, lognormal, normal, Pareto, 551 Student's t, triangular, uniform, and Weibull. 552 </p></blockquote></div> 553<p> 554 It calculates pdf, cdf, survivor, log survivor, hazard, tail areas, & critical 555 values for 5 tail values. 556 </p> 557<p> 558 It is also the only independent source found for the Weibull distribution; 559 unfortunately it appears to suffer from very poor accuracy in areas where the 560 underlying special function is known to be difficult to implement. 561 </p> 562<h5> 563<a name="math_toolkit.sf_implementation.h13"></a> 564 <span class="phrase"><a name="math_toolkit.sf_implementation.testing_for_invalid_parameters_t"></a></span><a class="link" href="sf_implementation.html#math_toolkit.sf_implementation.testing_for_invalid_parameters_t">Testing 565 for Invalid Parameters to Functions and Constructors</a> 566 </h5> 567<p> 568 After finding that some 'bad' parameters (like NaN) were not throwing a <code class="computeroutput"><span class="identifier">domain_error</span></code> exception as they should, a 569 function 570 </p> 571<p> 572 <code class="computeroutput"><span class="identifier">check_out_of_range</span></code> (in <code class="computeroutput"><span class="identifier">test_out_of_range</span><span class="special">.</span><span class="identifier">hpp</span></code>) was devised by JM to check (using Boost.Test's 573 BOOST_CHECK_THROW macro) that bad parameters passed to constructors and functions 574 throw <code class="computeroutput"><span class="identifier">domain_error</span></code> exceptions. 575 </p> 576<p> 577 Usage is <code class="computeroutput"><span class="identifier">check_out_of_range</span><span class="special"><</span> <span class="identifier">DistributionType</span> 578 <span class="special">>(</span><span class="identifier">list</span><span class="special">-</span><span class="identifier">of</span><span class="special">-</span><span class="identifier">params</span><span class="special">);</span></code> 579 Where list-of-params is a list of <span class="bold"><strong>valid</strong></span> parameters 580 from which the distribution can be constructed - ie the same number of args 581 are passed to the function, as are passed to the distribution constructor. 582 </p> 583<p> 584 The values of the parameters are not important, but must be <span class="bold"><strong>valid</strong></span> 585 to pass the constructor checks; the default values are suitable, but must be 586 explicitly provided, for example: 587 </p> 588<pre class="programlisting"><span class="identifier">check_out_of_range</span><span class="special"><</span><span class="identifier">extreme_value_distribution</span><span class="special"><</span><span class="identifier">RealType</span><span class="special">></span> <span class="special">>(</span><span class="number">1</span><span class="special">,</span> <span class="number">2</span><span class="special">);</span> 589</pre> 590<p> 591 Checks made are: 592 </p> 593<div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: disc; "> 594<li class="listitem"> 595 Infinity or NaN (if available) passed in place of each of the valid params. 596 </li> 597<li class="listitem"> 598 Infinity or NaN (if available) as a random variable. 599 </li> 600<li class="listitem"> 601 Out-of-range random variable passed to pdf and cdf (ie outside of "range(DistributionType)"). 602 </li> 603<li class="listitem"> 604 Out-of-range probability passed to quantile function and complement. 605 </li> 606</ul></div> 607<p> 608 but does <span class="bold"><strong>not</strong></span> check finite but out-of-range 609 parameters to the constructor because these are specific to each distribution, 610 for example: 611 </p> 612<pre class="programlisting"><span class="identifier">BOOST_CHECK_THROW</span><span class="special">(</span><span class="identifier">pdf</span><span class="special">(</span><span class="identifier">pareto_distribution</span><span class="special"><</span><span class="identifier">RealType</span><span class="special">>(</span><span class="number">0</span><span class="special">,</span> <span class="number">1</span><span class="special">),</span> <span class="number">0</span><span class="special">),</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">domain_error</span><span class="special">);</span> 613<span class="identifier">BOOST_CHECK_THROW</span><span class="special">(</span><span class="identifier">pdf</span><span class="special">(</span><span class="identifier">pareto_distribution</span><span class="special"><</span><span class="identifier">RealType</span><span class="special">>(</span><span class="number">1</span><span class="special">,</span> <span class="number">0</span><span class="special">),</span> <span class="number">0</span><span class="special">),</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">domain_error</span><span class="special">);</span> 614</pre> 615<p> 616 checks <code class="computeroutput"><span class="identifier">scale</span></code> and <code class="computeroutput"><span class="identifier">shape</span></code> parameters are both > 0 by checking 617 that <code class="computeroutput"><span class="identifier">domain_error</span></code> exception 618 is thrown if either are == 0. 619 </p> 620<p> 621 (Use of <code class="computeroutput"><span class="identifier">check_out_of_range</span></code> 622 function may mean that some previous tests are now redundant). 623 </p> 624<p> 625 It was also noted that if more than one parameter is bad, then only the first 626 detected will be reported by the error message. 627 </p> 628<h5> 629<a name="math_toolkit.sf_implementation.h14"></a> 630 <span class="phrase"><a name="math_toolkit.sf_implementation.creating_and_managing_the_equati"></a></span><a class="link" href="sf_implementation.html#math_toolkit.sf_implementation.creating_and_managing_the_equati">Creating 631 and Managing the Equations</a> 632 </h5> 633<p> 634 Equations that fit on a single line can most easily be produced by inline Quickbook 635 code using templates for Unicode Greek and Unicode Math symbols. All Greek 636 letter and small set of Math symbols is available at /boost-path/libs/math/doc/sf_and_dist/html4_symbols.qbk 637 </p> 638<p> 639 Where equations need to use more than one line, real Math editors were used. 640 </p> 641<p> 642 The primary source for the equations is now <a href="http://www.w3.org/Math/" target="_top">MathML</a>: 643 see the *.mml files in libs/math/doc/sf_and_dist/equations/. 644 </p> 645<p> 646 These are most easily edited by a GUI editor such as <a href="http://mathcast.sourceforge.net/home.html" target="_top">Mathcast</a>, 647 please note that the equation editor supplied with Open Office currently mangles 648 these files and should not currently be used. 649 </p> 650<p> 651 Conversion to SVG was achieved using <a href="https://sourceforge.net/projects/svgmath/" target="_top">SVGMath</a> 652 and a command line such as: 653 </p> 654<pre class="programlisting">$for file in *.mml; do 655>/cygdrive/c/Python25/python.exe 'C:\download\open\SVGMath-0.3.1\math2svg.py' \ 656>>$file > $(basename $file .mml).svg 657>done 658</pre> 659<p> 660 See also the section on "Using Python to run Inkscape" and "Using 661 inkscape to convert scalable vector SVG files to Portable Network graphic PNG". 662 </p> 663<p> 664 Note that SVGMath requires that the mml files are <span class="bold"><strong>not</strong></span> 665 wrapped in an XHTML XML wrapper - this is added by Mathcast by default - one 666 workaround is to copy an existing mml file and then edit it with Mathcast: 667 the existing format should then be preserved. This is a bug in the XML parser 668 used by SVGMath which the author is aware of. 669 </p> 670<p> 671 If necessary the XHTML wrapper can be removed with: 672 </p> 673<pre class="programlisting">cat filename | tr -d "\r\n" | sed -e 's/.*\(<math[^>]*>.*</math>\).*/\1/' > newfile</pre> 674<p> 675 Setting up fonts for SVGMath is currently rather tricky, on a Windows XP system 676 JM's font setup is the same as the sample config file provided with SVGMath 677 but with: 678 </p> 679<pre class="programlisting"> <!-- Double-struck --> 680 <mathvariant name="double-struck" family="Mathematica7, Lucida Sans Unicode"/> 681</pre> 682<p> 683 changed to: 684 </p> 685<pre class="programlisting"> <!-- Double-struck --> 686 <mathvariant name="double-struck" family="Lucida Sans Unicode"/> 687</pre> 688<p> 689 Note that unlike the sample config file supplied with SVGMath, this does not 690 make use of the <a href="http://support.wolfram.com/technotes/fonts/windows/latestfonts.html" target="_top">Mathematica 691 7 font</a> as this lacks sufficient Unicode information for it to be used 692 with either SVGMath or XEP "as is". 693 </p> 694<p> 695 Also note that the SVG files in the repository are almost certainly Windows-specific 696 since they reference various Windows Fonts. 697 </p> 698<p> 699 PNG files can be created from the SVGs using <a href="http://xmlgraphics.apache.org/batik/tools/rasterizer.html" target="_top">Batik</a> 700 and a command such as: 701 </p> 702<pre class="programlisting">java -jar 'C:\download\open\batik-1.7\batik-rasterizer.jar' -dpi 120 *.svg</pre> 703<p> 704 Or using Inkscape (File, Export bitmap, Drawing tab, bitmap size (default size, 705 100 dpi), Filename (default). png) 706 </p> 707<p> 708 or Using Cygwin, a command such as: 709 </p> 710<pre class="programlisting">for file in *.svg; do 711 /cygdrive/c/progra~1/Inkscape/inkscape -d 120 -e $(cygpath -a -w $(basename $file .svg).png) $(cygpath -a -w $file); 712done</pre> 713<p> 714 Using BASH 715 </p> 716<pre class="programlisting"># Convert single SVG to PNG file. 717# /c/progra~1/Inkscape/inkscape -d 120 -e a.png a.svg 718</pre> 719<p> 720 or to convert All files in folder SVG to PNG. 721 </p> 722<pre class="programlisting">for file in *.svg; do 723/c/progra~1/Inkscape/inkscape -d 120 -e $(basename $file .svg).png $file 724done 725</pre> 726<p> 727 Currently Inkscape seems to generate the better looking PNGs. 728 </p> 729<p> 730 The PDF is generated into \pdf\math.pdf using a command from a shell or command 731 window with current directory \math_toolkit\libs\math\doc\sf_and_dist, typically: 732 </p> 733<pre class="programlisting">bjam -a pdf >math_pdf.log</pre> 734<p> 735 Note that XEP will have to be configured to <span class="bold"><strong>use and embed</strong></span> 736 whatever fonts are used by the SVG equations (almost certainly editing the 737 sample xep.xml provided by the XEP installation). If you fail to do this you 738 will get XEP warnings in the log file like 739 </p> 740<pre class="programlisting">[warning]could not find any font family matching "Times New Roman"; replaced by Helvetica</pre> 741<p> 742 (html is the default so it is generated at libs\math\doc\html\index.html using 743 command line >bjam -a > math_toolkit.docs.log). 744 </p> 745<pre class="programlisting"><span class="special"><!--</span> <span class="identifier">Sample</span> <span class="identifier">configuration</span> <span class="keyword">for</span> <span class="identifier">Windows</span> <span class="identifier">TrueType</span> <span class="identifier">fonts</span><span class="special">.</span> <span class="special">--></span> 746</pre> 747<p> 748 is provided in the xep.xml downloaded, but the Windows TrueType fonts are commented 749 out. 750 </p> 751<p> 752 JM's XEP config file \xep\xep.xml has the following font configuration section 753 added: 754 </p> 755<pre class="programlisting"> <font-group xml:base="file:/C:/Windows/Fonts/" label="Windows TrueType" embed="true" subset="true"> 756 <font-family name="Arial"> 757 <font><font-data ttf="arial.ttf"/></font> 758 <font style="oblique"><font-data ttf="ariali.ttf"/></font> 759 <font weight="bold"><font-data ttf="arialbd.ttf"/></font> 760 <font weight="bold" style="oblique"><font-data ttf="arialbi.ttf"/></font> 761 </font-family> 762 763 <font-family name="Times New Roman" ligatures="&#xFB01; &#xFB02;"> 764 <font><font-data ttf="times.ttf"/></font> 765 <font style="italic"><font-data ttf="timesi.ttf"/></font> 766 <font weight="bold"><font-data ttf="timesbd.ttf"/></font> 767 <font weight="bold" style="italic"><font-data ttf="timesbi.ttf"/></font> 768 </font-family> 769 770 <font-family name="Courier New"> 771 <font><font-data ttf="cour.ttf"/></font> 772 <font style="oblique"><font-data ttf="couri.ttf"/></font> 773 <font weight="bold"><font-data ttf="courbd.ttf"/></font> 774 <font weight="bold" style="oblique"><font-data ttf="courbi.ttf"/></font> 775 </font-family> 776 777 <font-family name="Tahoma" embed="true"> 778 <font><font-data ttf="tahoma.ttf"/></font> 779 <font weight="bold"><font-data ttf="tahomabd.ttf"/></font> 780 </font-family> 781 782 <font-family name="Verdana" embed="true"> 783 <font><font-data ttf="verdana.ttf"/></font> 784 <font style="oblique"><font-data ttf="verdanai.ttf"/></font> 785 <font weight="bold"><font-data ttf="verdanab.ttf"/></font> 786 <font weight="bold" style="oblique"><font-data ttf="verdanaz.ttf"/></font> 787 </font-family> 788 789 <font-family name="Palatino" embed="true" ligatures="&#xFB00; &#xFB01; &#xFB02; &#xFB03; &#xFB04;"> 790 <font><font-data ttf="pala.ttf"/></font> 791 <font style="italic"><font-data ttf="palai.ttf"/></font> 792 <font weight="bold"><font-data ttf="palab.ttf"/></font> 793 <font weight="bold" style="italic"><font-data ttf="palabi.ttf"/></font> 794 </font-family> 795 796 <font-family name="Lucida Sans Unicode"> 797 <!-- <font><font-data ttf="lsansuni.ttf"><<span class="emphasis"><em>font> --> 798 <!-- actually called l_10646.ttf on Windows 2000 and Vista Sp1 --> 799 <font><font-data ttf="l_10646.ttf"</em></span>></font> 800 </font-family> 801</pre> 802<p> 803 PAB had to alter his because the Lucida Sans Unicode font had a different name. 804 Other changes are very likely to be required if you are not using Windows. 805 </p> 806<p> 807 XZ authored his equations using the venerable Latex, JM converted these to 808 MathML using <a href="http://gentoo-wiki.com/HOWTO_Convert_LaTeX_to_HTML_with_MathML" target="_top">mxlatex</a>. 809 This process is currently unreliable and required some manual intervention: 810 consequently Latex source is not considered a viable route for the automatic 811 production of SVG versions of equations. 812 </p> 813<p> 814 Equations are embedded in the quickbook source using the <span class="emphasis"><em>equation</em></span> 815 template defined in math.qbk. This outputs Docbook XML that looks like: 816 </p> 817<pre class="programlisting"><inlinemediaobject> 818<imageobject role="html"> 819<imagedata fileref="../equations/myfile.png"></imagedata> 820</imageobject> 821<imageobject role="print"> 822<imagedata fileref="../equations/myfile.svg"></imagedata> 823</imageobject> 824</inlinemediaobject> 825</pre> 826<p> 827 MathML is not currently present in the Docbook output, or in the generated 828 HTML: this needs further investigation. 829 </p> 830<h5> 831<a name="math_toolkit.sf_implementation.h15"></a> 832 <span class="phrase"><a name="math_toolkit.sf_implementation.producing_graphs"></a></span><a class="link" href="sf_implementation.html#math_toolkit.sf_implementation.producing_graphs">Producing 833 Graphs</a> 834 </h5> 835<p> 836 Graphs were produced in SVG format and then converted to PNG's using the same 837 process as the equations. 838 </p> 839<p> 840 The programs <code class="computeroutput"><span class="special">/</span><span class="identifier">libs</span><span class="special">/</span><span class="identifier">math</span><span class="special">/</span><span class="identifier">doc</span><span class="special">/</span><span class="identifier">sf_and_dist</span><span class="special">/</span><span class="identifier">graphs</span><span class="special">/</span><span class="identifier">dist_graphs</span><span class="special">.</span><span class="identifier">cpp</span></code> and <code class="computeroutput"><span class="special">/</span><span class="identifier">libs</span><span class="special">/</span><span class="identifier">math</span><span class="special">/</span><span class="identifier">doc</span><span class="special">/</span><span class="identifier">sf_and_dist</span><span class="special">/</span><span class="identifier">graphs</span><span class="special">/</span><span class="identifier">sf_graphs</span><span class="special">.</span><span class="identifier">cpp</span></code> generate 841 the SVG's directly using the <a href="http://code.google.com/soc/2007/boost/about.html" target="_top">Google 842 Summer of Code 2007</a> project of Jacob Voytko (whose work so far, considerably 843 enhanced and now reasonably mature and usable, by Paul A. Bristow, is at .\boost-sandbox\SOC\2007\visualization). 844 </p> 845</div> 846<table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr> 847<td align="left"></td> 848<td align="right"><div class="copyright-footer">Copyright © 2006-2019 Nikhar 849 Agrawal, Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos, 850 Hubert Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Matthew Pulver, Johan 851 Råde, Gautam Sewani, Benjamin Sobotta, Nicholas Thompson, Thijs van den Berg, 852 Daryle Walker and Xiaogang Zhang<p> 853 Distributed under the Boost Software License, Version 1.0. (See accompanying 854 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>) 855 </p> 856</div></td> 857</tr></table> 858<hr> 859<div class="spirit-nav"> 860<a accesskey="p" href="../backgrounders.html"><img src="../../../../../doc/src/images/prev.png" alt="Prev"></a><a accesskey="u" href="../backgrounders.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="special_tut.html"><img src="../../../../../doc/src/images/next.png" alt="Next"></a> 861</div> 862</body> 863</html> 864
