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| <div class="section"> |
| <div class="titlepage"><div><div><h2 class="title" style="clear: both"> |
| <a name="math_toolkit.constants_faq"></a><a class="link" href="constants_faq.html" title="FAQs">FAQs</a> |
| </h2></div></div></div> |
| <h5> |
| <a name="math_toolkit.constants_faq.h0"></a> |
| <span class="phrase"><a name="math_toolkit.constants_faq.why_are_these_constants_chosen"></a></span><a class="link" href="constants_faq.html#math_toolkit.constants_faq.why_are_these_constants_chosen">Why are |
| <span class="emphasis"><em>these</em></span> Constants Chosen?</a> |
| </h5> |
| <p> |
| It is, of course, impossible to please everyone with a list like this. |
| </p> |
| <p> |
| Some of the criteria we have used are: |
| </p> |
| <div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: disc; "> |
| <li class="listitem"> |
| Used in Boost.Math. |
| </li> |
| <li class="listitem"> |
| Commonly used. |
| </li> |
| <li class="listitem"> |
| Expensive to compute. |
| </li> |
| <li class="listitem"> |
| Requested by users. |
| </li> |
| <li class="listitem"> |
| <a href="http://en.wikipedia.org/wiki/Mathematical_constant" target="_top">Used in |
| science and mathematics.</a> |
| </li> |
| <li class="listitem"> |
| No integer values (because so cheap to construct).<br> (You can easily |
| define your own if found convenient, for example: <code class="computeroutput"><span class="identifier">FPT</span> |
| <span class="identifier">one</span> <span class="special">=</span><span class="keyword">static_cast</span><span class="special"><</span><span class="identifier">FPT</span><span class="special">>(</span><span class="number">42</span><span class="special">);</span></code>). |
| </li> |
| </ul></div> |
| <h5> |
| <a name="math_toolkit.constants_faq.h1"></a> |
| <span class="phrase"><a name="math_toolkit.constants_faq.how_are_constants_named"></a></span><a class="link" href="constants_faq.html#math_toolkit.constants_faq.how_are_constants_named">How |
| are constants named?</a> |
| </h5> |
| <div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: disc; "> |
| <li class="listitem"> |
| Not macros, so no upper case. |
| </li> |
| <li class="listitem"> |
| All lower case (following C++ standard names). |
| </li> |
| <li class="listitem"> |
| No CamelCase. |
| </li> |
| <li class="listitem"> |
| Underscore as _ delimiter between words. |
| </li> |
| <li class="listitem"> |
| Numbers spelt as words rather than decimal digits (except following pow). |
| </li> |
| <li class="listitem"> |
| Abbreviation conventions: |
| <div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: circle; "> |
| <li class="listitem"> |
| root for square root. |
| </li> |
| <li class="listitem"> |
| cbrt for cube root. |
| </li> |
| <li class="listitem"> |
| pow for pow function using decimal digits like pow23 for n<sup>2/3</sup>. |
| </li> |
| <li class="listitem"> |
| div for divided by or operator /. |
| </li> |
| <li class="listitem"> |
| minus for operator -, plus for operator +. |
| </li> |
| <li class="listitem"> |
| sqr for squared. |
| </li> |
| <li class="listitem"> |
| cubed for cubed n<sup>3</sup>. |
| </li> |
| <li class="listitem"> |
| words for greek, like π, ζ and Γ. |
| </li> |
| <li class="listitem"> |
| words like half, third, three_quarters, sixth for fractions. (Digit(s) |
| can get muddled). |
| </li> |
| <li class="listitem"> |
| log10 for log<sub>10</sub> |
| </li> |
| <li class="listitem"> |
| ln for log<sub>e</sub> |
| </li> |
| </ul></div> |
| </li> |
| </ul></div> |
| <h5> |
| <a name="math_toolkit.constants_faq.h2"></a> |
| <span class="phrase"><a name="math_toolkit.constants_faq.how_are_the_constants_derived"></a></span><a class="link" href="constants_faq.html#math_toolkit.constants_faq.how_are_the_constants_derived">How are |
| the constants derived?</a> |
| </h5> |
| <p> |
| The constants have all been calculated using high-precision software working |
| with up to 300-bit precision giving about 100 decimal digits. (The precision |
| can be arbitrarily chosen and is limited only by compute time). |
| </p> |
| <h5> |
| <a name="math_toolkit.constants_faq.h3"></a> |
| <span class="phrase"><a name="math_toolkit.constants_faq.how_accurate_are_the_constants"></a></span><a class="link" href="constants_faq.html#math_toolkit.constants_faq.how_accurate_are_the_constants">How Accurate |
| are the constants?</a> |
| </h5> |
| <p> |
| The minimum accuracy chosen (100 decimal digits) exceeds the accuracy of reasonably-foreseeable |
| floating-point hardware (256-bit) and should meet most high-precision computations. |
| </p> |
| <h5> |
| <a name="math_toolkit.constants_faq.h4"></a> |
| <span class="phrase"><a name="math_toolkit.constants_faq.how_are_the_constants_tested"></a></span><a class="link" href="constants_faq.html#math_toolkit.constants_faq.how_are_the_constants_tested">How are the |
| constants tested?</a> |
| </h5> |
| <div class="orderedlist"><ol class="orderedlist" type="1"> |
| <li class="listitem"> |
| Comparison using Boost.Test BOOST_CHECK_CLOSE_FRACTION using long double |
| literals, with at least 35 decimal digits, enough to be accurate for all |
| long double implementations. The tolerance is usually twice <code class="computeroutput"><span class="keyword">long</span> <span class="keyword">double</span> <span class="identifier">epsilon</span></code>. |
| </li> |
| <li class="listitem"> |
| Comparison with calculation at long double precision. This often requires |
| a slightly higher tolerance than two epsilon because of computational noise |
| from round-off etc, especially when trig and other functions are called. |
| </li> |
| <li class="listitem"> |
| Comparison with independent published values, for example, using <a href="http://oeis.org/" target="_top">The On-Line Encyclopedia of Integer Sequences (OEIS)</a> |
| again using at least 35 decimal digits strings. |
| </li> |
| <li class="listitem"> |
| Comparison with independely calculated values using arbitrary precision |
| tools like <a href="http://www.wolfram.com/mathematica/" target="_top">Mathematica</a>, |
| again using at least 35 decimal digits literal strings. |
| </li> |
| </ol></div> |
| <div class="warning"><table border="0" summary="Warning"> |
| <tr> |
| <td rowspan="2" align="center" valign="top" width="25"><img alt="[Warning]" src="../../../../../doc/src/images/warning.png"></td> |
| <th align="left">Warning</th> |
| </tr> |
| <tr><td align="left" valign="top"><p> |
| We have not yet been able to <span class="bold"><strong>check</strong></span> that |
| <span class="bold"><strong>all</strong></span> constants are accurate at the full arbitrary |
| precision, at present 100 decimal digits. But certain key values like <code class="computeroutput"><span class="identifier">e</span></code> and <code class="computeroutput"><span class="identifier">pi</span></code> |
| appear to be accurate and internal consistencies suggest that others are |
| this accurate too. |
| </p></td></tr> |
| </table></div> |
| <h5> |
| <a name="math_toolkit.constants_faq.h5"></a> |
| <span class="phrase"><a name="math_toolkit.constants_faq.why_is_portability_important"></a></span><a class="link" href="constants_faq.html#math_toolkit.constants_faq.why_is_portability_important">Why is Portability |
| important?</a> |
| </h5> |
| <p> |
| Code written using math constants is easily portable even when using different |
| floating-point types with differing precision. |
| </p> |
| <p> |
| It is a mistake to expect that results of computations will be <span class="bold"><strong>identical</strong></span>, |
| but you can achieve the <span class="bold"><strong>best accuracy possible for the |
| floating-point type in use</strong></span>. |
| </p> |
| <p> |
| This has no extra cost to the user, but reduces irritating, and often confusing |
| and very hard-to-trace effects, caused by the intrinsically limited precision |
| of floating-point calculations. |
| </p> |
| <p> |
| A harmless symptom of this limit is a spurious least-significant digit; at |
| worst, slightly inaccurate constants sometimes cause iterating algorithms to |
| diverge wildly because internal comparisons just fail. |
| </p> |
| <h5> |
| <a name="math_toolkit.constants_faq.h6"></a> |
| <span class="phrase"><a name="math_toolkit.constants_faq.what_is_the_internal_format_of_t"></a></span><a class="link" href="constants_faq.html#math_toolkit.constants_faq.what_is_the_internal_format_of_t">What |
| is the Internal Format of the constants, and why?</a> |
| </h5> |
| <p> |
| See <a class="link" href="tutorial.html" title="Tutorial">tutorial</a> above for normal |
| use, but this FAQ explains the internal details used for the constants. |
| </p> |
| <p> |
| Constants are stored as 100 decimal digit values. However, some compilers do |
| not accept decimal digits strings as long as this. So the constant is split |
| into two parts, with the first containing at least 128-bit long double precision |
| (35 decimal digits), and for consistency should be in scientific format with |
| a signed exponent. |
| </p> |
| <p> |
| The second part is the value of the constant expressed as a string literal, |
| accurate to at least 100 decimal digits (in practice that means at least 102 |
| digits). Again for consistency use scientific format with a signed exponent. |
| </p> |
| <p> |
| For types with precision greater than a long double, then if T is constructible |
| <code class="computeroutput"><span class="identifier">T</span> </code>is constructible from a |
| <code class="computeroutput"><span class="keyword">const</span> <span class="keyword">char</span><span class="special">*</span></code> then it's directly constructed from the string, |
| otherwise we fall back on lexical_cast to convert to type <code class="computeroutput"><span class="identifier">T</span></code>. |
| (Using a string is necessary because you can't use a numeric constant since |
| even a <code class="computeroutput"><span class="keyword">long</span> <span class="keyword">double</span></code> |
| might not have enough digits). |
| </p> |
| <p> |
| So, for example, a constant like pi is internally defined as |
| </p> |
| <pre class="programlisting"><span class="identifier">BOOST_DEFINE_MATH_CONSTANT</span><span class="special">(</span><span class="identifier">pi</span><span class="special">,</span> <span class="number">3.141592653589793238462643383279502884e+00</span><span class="special">,</span> <span class="string">"3.14159265358979323846264338327950288419716939937510582097494459230781640628620899862803482534211706798214808651e+00"</span><span class="special">);</span> |
| </pre> |
| <p> |
| In this case the significand is 109 decimal digits, ensuring 100 decimal digits |
| are exact, and exponent is zero. |
| </p> |
| <p> |
| See <a class="link" href="new_const.html" title="Defining New Constants">defining new constants</a> to |
| calculate new constants. |
| </p> |
| <p> |
| A macro definition like this can be pasted into user code where convenient, |
| or into <code class="computeroutput"><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">hpp</span></code> if it |
| is to be added to the Boost.Math library. |
| </p> |
| <h5> |
| <a name="math_toolkit.constants_faq.h7"></a> |
| <span class="phrase"><a name="math_toolkit.constants_faq.what_floating_point_types_could_"></a></span><a class="link" href="constants_faq.html#math_toolkit.constants_faq.what_floating_point_types_could_">What |
| Floating-point Types could I use?</a> |
| </h5> |
| <p> |
| Apart from the built-in floating-point types <code class="computeroutput"><span class="keyword">float</span></code>, |
| <code class="computeroutput"><span class="keyword">double</span></code>, <code class="computeroutput"><span class="keyword">long</span> |
| <span class="keyword">double</span></code>, there are several arbitrary |
| precision floating-point classes available, but most are not licensed for commercial |
| use. |
| </p> |
| <h6> |
| <a name="math_toolkit.constants_faq.h8"></a> |
| <span class="phrase"><a name="math_toolkit.constants_faq.boost_multiprecision_by_christop"></a></span><a class="link" href="constants_faq.html#math_toolkit.constants_faq.boost_multiprecision_by_christop">Boost.Multiprecision |
| by Christopher Kormanyos</a> |
| </h6> |
| <p> |
| This work is based on an earlier work called e-float: Algorithm 910: A Portable |
| C++ Multiple-Precision System for Special-Function Calculations, in ACM TOMS, |
| {VOL 37, ISSUE 4, (February 2011)} (C) ACM, 2011. <a href="http://doi.acm.org/10.1145/1916461.1916469" target="_top">http://doi.acm.org/10.1145/1916461.1916469</a> |
| <a href="https://svn.boost.org/svn/boost/sandbox/e_float/" target="_top">e_float</a> |
| but is now re-factored and available under the Boost license in the Boost-sandbox |
| at <a href="https://svn.boost.org/svn/boost/sandbox/multiprecision/" target="_top">multiprecision</a> |
| where it is being refined and prepared for review. |
| </p> |
| <h6> |
| <a name="math_toolkit.constants_faq.h9"></a> |
| <span class="phrase"><a name="math_toolkit.constants_faq.boost_cpp_float_by_john_maddock_"></a></span><a class="link" href="constants_faq.html#math_toolkit.constants_faq.boost_cpp_float_by_john_maddock_">Boost.cpp_float |
| by John Maddock using Expression Templates</a> |
| </h6> |
| <p> |
| <a href="https://svn.boost.org/svn/boost/sandbox/big_number/" target="_top">Big Number</a> |
| which is a reworking of <a href="https://svn.boost.org/svn/boost/sandbox/e_float/" target="_top">e_float</a> |
| by Christopher Kormanyos to use expression templates for faster execution. |
| </p> |
| <h6> |
| <a name="math_toolkit.constants_faq.h10"></a> |
| <span class="phrase"><a name="math_toolkit.constants_faq.ntl_class_quad_float"></a></span><a class="link" href="constants_faq.html#math_toolkit.constants_faq.ntl_class_quad_float">NTL |
| class quad_float</a> |
| </h6> |
| <p> |
| <a href="http://shoup.net/ntl/" target="_top">NTL</a> by Victor Shoup has fixed and |
| arbitrary high precision fixed and floating-point types. However none of these |
| are licenced for commercial use. |
| </p> |
| <pre class="programlisting"><span class="preprocessor">#include</span> <span class="special"><</span><span class="identifier">NTL</span><span class="special">/</span><span class="identifier">quad_float</span><span class="special">.</span><span class="identifier">h</span><span class="special">></span> <span class="comment">// quad precision 106-bit, about 32 decimal digits.</span> |
| <span class="keyword">using</span> <span class="identifier">NTL</span><span class="special">::</span><span class="identifier">to_quad_float</span><span class="special">;</span> <span class="comment">// Less precise than arbitrary precision NTL::RR.</span> |
| </pre> |
| <p> |
| NTL class <code class="computeroutput"><span class="identifier">quad_float</span></code>, which |
| gives a form of quadruple precision, 106-bit significand (but without an extended |
| exponent range.) With an IEC559/IEEE 754 compatible processor, for example |
| Intel X86 family, with 64-bit double, and 53-bit significand, using the significands |
| of <span class="bold"><strong>two</strong></span> 64-bit doubles, 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="keyword">double</span><span class="special">>::</span><span class="identifier">digits10</span></code> is 16, then we get about twice the |
| precision, so <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">quad_float</span><span class="special">>::</span><span class="identifier">digits10</span><span class="special">()</span></code> |
| should be 32. (the default <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">RR</span><span class="special">>::</span><span class="identifier">digits10</span><span class="special">()</span></code> |
| should be about 40). (which seems to agree with experiments). We output constants |
| (including some noisy bits, an approximation to <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">RR</span><span class="special">>::</span><span class="identifier">max_digits10</span><span class="special">()</span></code>) |
| by adding 2 extra decimal digits, so using <code class="computeroutput"><span class="identifier">quad_float</span><span class="special">::</span><span class="identifier">SetOutputPrecision</span><span class="special">(</span><span class="number">32</span> <span class="special">+</span> |
| <span class="number">2</span><span class="special">);</span></code> |
| </p> |
| <p> |
| Apple Mac/Darwin uses a similar <span class="emphasis"><em>doubledouble</em></span> 106-bit for |
| its built-in <code class="computeroutput"><span class="keyword">long</span> <span class="keyword">double</span></code> |
| type. |
| </p> |
| <div class="note"><table border="0" summary="Note"> |
| <tr> |
| <td rowspan="2" align="center" valign="top" width="25"><img alt="[Note]" src="../../../../../doc/src/images/note.png"></td> |
| <th align="left">Note</th> |
| </tr> |
| <tr><td align="left" valign="top"><p> |
| The precision of all <code class="computeroutput"><span class="identifier">doubledouble</span></code> |
| floating-point types is rather odd and values given are only approximate. |
| </p></td></tr> |
| </table></div> |
| <p> |
| <span class="bold"><strong>New projects should use <a href="http://www.boost.org/doc/libs/1_53_0_beta1/libs/multiprecision/doc/html/index.html" target="_top">Boost.Multiprecision</a>.</strong></span> |
| </p> |
| <h6> |
| <a name="math_toolkit.constants_faq.h11"></a> |
| <span class="phrase"><a name="math_toolkit.constants_faq.ntl_class_rr"></a></span><a class="link" href="constants_faq.html#math_toolkit.constants_faq.ntl_class_rr">NTL |
| class RR</a> |
| </h6> |
| <p> |
| Arbitrary precision floating point with NTL class RR, default is 150 bit (about |
| 50 decimal digits) used here with 300 bit to output 100 decimal digits, enough |
| for many practical non-'number-theoretic' C++ applications. |
| </p> |
| <p> |
| <a href="http://www.shoup.net/ntl/" target="_top">NTL A Library for doing Number Theory</a> |
| is <span class="bold"><strong>not licenced for commercial use</strong></span>. |
| </p> |
| <p> |
| This class is used in Boost.Math and is an option when using big_number projects |
| to calculate new math constants. |
| </p> |
| <p> |
| <span class="bold"><strong>New projects should use <a href="http://www.boost.org/doc/libs/1_53_0_beta1/libs/multiprecision/doc/html/index.html" target="_top">Boost.Multiprecision</a>.</strong></span> |
| </p> |
| <h6> |
| <a name="math_toolkit.constants_faq.h12"></a> |
| <span class="phrase"><a name="math_toolkit.constants_faq.gmp_and_mpfr"></a></span><a class="link" href="constants_faq.html#math_toolkit.constants_faq.gmp_and_mpfr">GMP |
| and MPFR</a> |
| </h6> |
| <p> |
| <a href="http://gmplib.org" target="_top">GMP</a> and <a href="http://www.mpfr.org/" target="_top">MPFR</a> |
| have also been used to compute constants, but are licensed under the <a href="http://www.gnu.org/copyleft/lesser.html" target="_top">Lesser GPL license</a> and |
| are <span class="bold"><strong>not licensed for commercial use</strong></span>. |
| </p> |
| <h5> |
| <a name="math_toolkit.constants_faq.h13"></a> |
| <span class="phrase"><a name="math_toolkit.constants_faq.what_happened_to_a_previous_coll"></a></span><a class="link" href="constants_faq.html#math_toolkit.constants_faq.what_happened_to_a_previous_coll">What |
| happened to a previous collection of constants proposed for Boost?</a> |
| </h5> |
| <p> |
| A review concluded that the way in which the constants were presented did not |
| meet many peoples needs. None of the methods proposed met many users' essential |
| requirement to allow writing simply <code class="computeroutput"><span class="identifier">pi</span></code> |
| rather than <code class="computeroutput"><span class="identifier">pi</span><span class="special">()</span></code>. |
| Many science and engineering equations look difficult to read when because |
| function call brackets can be confused with the many other brackets often needed. |
| All the methods then proposed of avoiding the brackets failed to meet all needs, |
| often on grounds of complexity and lack of applicability to various realistic |
| scenarios. |
| </p> |
| <p> |
| So the simple namespace method, proposed on its own, but rejected at the first |
| review, has been added to allow users to have convenient access to float, double |
| and long double values, but combined with template struct and functions to |
| allow simultaneous use with other non-built-in floating-point types. |
| </p> |
| <h5> |
| <a name="math_toolkit.constants_faq.h14"></a> |
| <span class="phrase"><a name="math_toolkit.constants_faq.why_do_the_constants_internally_"></a></span><a class="link" href="constants_faq.html#math_toolkit.constants_faq.why_do_the_constants_internally_">Why do |
| the constants (internally) have a struct rather than a simple function?</a> |
| </h5> |
| <p> |
| A function mechanism was provided by in previous versions of Boost.Math. |
| </p> |
| <p> |
| The new mechanism is to permit partial specialization. See Custom Specializing |
| a constant above. It should also allow use with other packages like <a href="http://www.ttmath.org/" target="_top">ttmath Bignum C++ library.</a> |
| </p> |
| <h5> |
| <a name="math_toolkit.constants_faq.h15"></a> |
| <span class="phrase"><a name="math_toolkit.constants_faq.where_can_i_find_other_high_prec"></a></span><a class="link" href="constants_faq.html#math_toolkit.constants_faq.where_can_i_find_other_high_prec">Where |
| can I find other high precision constants?</a> |
| </h5> |
| <div class="orderedlist"><ol class="orderedlist" type="1"> |
| <li class="listitem"> |
| Constants with very high precision and good accuracy (>40 decimal digits) |
| from Simon Plouffe's web based collection <a href="http://pi.lacim.uqam.ca/eng/" target="_top">http://pi.lacim.uqam.ca/eng/</a>. |
| </li> |
| <li class="listitem"> |
| <a href="https://oeis.org/" target="_top">The On-Line Encyclopedia of Integer Sequences |
| (OEIS)</a> |
| </li> |
| <li class="listitem"> |
| Checks using printed text optically scanned values and converted from: |
| D. E. Knuth, Art of Computer Programming, Appendix A, Table 1, Vol 1, ISBN |
| 0 201 89683 4 (1997) |
| </li> |
| <li class="listitem"> |
| M. Abrahamovitz & I. E. Stegun, National Bureau of Standards, Handbook |
| of Mathematical Functions, a reference source for formulae now superceded |
| by |
| </li> |
| <li class="listitem"> |
| Frank W. Olver, Daniel W. Lozier, Ronald F. Boisvert, Charles W. Clark, |
| NIST Handbook of Mathemetical Functions, Cambridge University Press, ISBN |
| 978-0-521-14063-8, 2010. |
| </li> |
| <li class="listitem"> |
| John F Hart, Computer Approximations, Kreiger (1978) ISBN 0 88275 642 7. |
| </li> |
| <li class="listitem"> |
| Some values from Cephes Mathematical Library, Stephen L. Moshier and CALC100 |
| 100 decimal digit Complex Variable Calculator Program, a DOS utility. |
| </li> |
| <li class="listitem"> |
| Xavier Gourdon, Pascal Sebah, 50 decimal digits constants at <a href="http://numbers.computation.free.fr/Constants/constants.html" target="_top">Number, |
| constants and computation</a>. |
| </li> |
| </ol></div> |
| <h5> |
| <a name="math_toolkit.constants_faq.h16"></a> |
| <span class="phrase"><a name="math_toolkit.constants_faq.where_are_physical_constants"></a></span><a class="link" href="constants_faq.html#math_toolkit.constants_faq.where_are_physical_constants">Where are |
| Physical Constants?</a> |
| </h5> |
| <p> |
| Not here in this Boost.Math collection, because physical constants: |
| </p> |
| <div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: disc; "> |
| <li class="listitem"> |
| Are measurements, not truely constants. |
| </li> |
| <li class="listitem"> |
| Are not truly constant and keeping changing as mensuration technology improves. |
| </li> |
| <li class="listitem"> |
| Have a instrinsic uncertainty. |
| </li> |
| <li class="listitem"> |
| Mathematical constants are stored and represented at varying precision, |
| but should never be inaccurate. |
| </li> |
| </ul></div> |
| <p> |
| Some physical constants may be available in Boost.Units. |
| </p> |
| </div> |
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| <td align="right"><div class="copyright-footer">Copyright © 2006-2010, 2012-2014 Nikhar Agrawal, |
| Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos, Hubert |
| Holin, Bruno Lalande, John Maddock, Johan Råde, Gautam Sewani, Benjamin Sobotta, |
| Thijs van den Berg, Daryle Walker and Xiaogang Zhang<p> |
| Distributed under the Boost Software License, Version 1.0. (See accompanying |
| 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>) |
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