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- </div>
- <div class="section">
- <div class="titlepage"><div><div><h2 class="title" style="clear: both">
- <a name="math_toolkit.sf_implementation"></a><a class="link" href="sf_implementation.html" title="Additional Implementation Notes">Additional Implementation
- Notes</a>
- </h2></div></div></div>
- <p>
- The majority of the implementation notes are included with the documentation
- of each function or distribution. The notes here are of a more general nature,
- and reflect more the general implementation philosophy used.
- </p>
- <h5>
- <a name="math_toolkit.sf_implementation.h0"></a>
- <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
- philosophy</a>
- </h5>
- <p>
- "First be right, then be fast."
- </p>
- <p>
- There will always be potential compromises to be made between speed and accuracy.
- It may be possible to find faster methods, particularly for certain limited
- ranges of arguments, but for most applications of math functions and distributions,
- we judge that speed is rarely as important as accuracy.
- </p>
- <p>
- So our priority is accuracy.
- </p>
- <p>
- To permit evaluation of accuracy of the special functions, production of extremely
- accurate tables of test values has received considerable effort.
- </p>
- <p>
- (It also required much CPU effort - there was some danger of molten plastic
- dripping from the bottom of JM's laptop, so instead, PAB's Dual-core desktop
- was kept 50% busy for <span class="bold"><strong>days</strong></span> calculating some
- tables of test values!)
- </p>
- <p>
- For a specific RealType, say <code class="computeroutput"><span class="keyword">float</span></code>
- or <code class="computeroutput"><span class="keyword">double</span></code>, it may be possible
- to find approximations for some functions that are simpler and thus faster,
- but less accurate (perhaps because there are no refining iterations, for example,
- when calculating inverse functions).
- </p>
- <p>
- If these prove accurate enough to be "fit for his purpose", then
- a user may substitute his custom specialization.
- </p>
- <p>
- For example, there are approximations dating back from times when computation
- was a <span class="bold"><strong>lot</strong></span> more expensive:
- </p>
- <p>
- H Goldberg and H Levine, Approximate formulas for percentage points and normalisation
- of t and chi squared, Ann. Math. Stat., 17(4), 216 - 225 (Dec 1946).
- </p>
- <p>
- A H Carter, Approximations to percentage points of the z-distribution, Biometrika
- 34(2), 352 - 358 (Dec 1947).
- </p>
- <p>
- These could still provide sufficient accuracy for some speed-critical applications.
- </p>
- <h5>
- <a name="math_toolkit.sf_implementation.h1"></a>
- <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
- and Representation of Test Values</a>
- </h5>
- <p>
- In order to be accurate enough for as many as possible real types, constant
- values are given to 50 decimal digits if available (though many sources proved
- only accurate near to 64-bit double precision). Values are specified as long
- double types by appending L, unless they are exactly representable, for example
- integers, or binary fractions like 0.125. This avoids the risk of loss of accuracy
- 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
- the appropriate RealType for spot tests.
- </p>
- <p>
- Functions that return constants values, like kurtosis for example, are written
- as
- </p>
- <p>
- <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>
- <span class="number">5</span><span class="special">;</span></code>
- </p>
- <p>
- to provide the most accurate value that the compiler can compute for the real
- type. (The denominator is an integer and so will be promoted exactly).
- </p>
- <p>
- So tests for one third, <span class="bold"><strong>not</strong></span> exactly representable
- with radix two floating-point, (should) use, for example:
- </p>
- <p>
- <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>
- <span class="number">3</span><span class="special">;</span></code>
- </p>
- <p>
- If a function is very sensitive to changes in input, specifying an inexact
- value as input (such as 0.1) can throw the result off by a noticeable amount:
- 0.1f is "wrong" by ~1e-7 for example (because 0.1 has no exact binary
- representation). That is why exact binary values - halves, quarters, and eighths
- 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>
- with <code class="computeroutput"><span class="identifier">b</span></code> a power of two (in order
- to ensure that the result is an exactly representable binary value).
- </p>
- <h5>
- <a name="math_toolkit.sf_implementation.h2"></a>
- <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
- of Tests</a>
- </h5>
- <p>
- The tolerances need to be set to the maximum of:
- </p>
- <div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: disc; ">
- <li class="listitem">
- Some epsilon value.
- </li>
- <li class="listitem">
- The accuracy of the data (often only near 64-bit double).
- </li>
- </ul></div>
- <p>
- Otherwise when long double has more digits than the test data, then no amount
- of tweaking an epsilon based tolerance will work.
- </p>
- <p>
- A common problem is when tolerances that are suitable for implementations like
- Microsoft VS.NET where double and long double are the same size: tests fail
- on other systems where long double is more accurate than double. Check first
- that the suffix L is present, and then that the tolerance is big enough.
- </p>
- <h5>
- <a name="math_toolkit.sf_implementation.h3"></a>
- <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
- Unsuitable Arguments</a>
- </h5>
- <p>
- In <a href="http://www.open-std.org/jtc1/sc22/wg21/docs/papers/2004/n1665.pdf" target="_top">Errors
- in Mathematical Special Functions</a>, J. Marraffino & M. Paterno it
- is proposed that signalling a domain error is mandatory when the argument would
- give an mathematically undefined result.
- </p>
- <div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: disc; "><li class="listitem">
- Guideline 1
- </li></ul></div>
- <div class="blockquote"><blockquote class="blockquote"><p>
- A mathematical function is said to be defined at a point a = (a1, a2, . .
- .) if the limits as x = (x1, x2, . . .) 'approaches a from all directions
- agree'. The defined value may be any number, or +infinity, or -infinity.
- </p></blockquote></div>
- <p>
- Put crudely, if the function goes to + infinity and then emerges 'round-the-back'
- with - infinity, it is NOT defined.
- </p>
- <div class="blockquote"><blockquote class="blockquote"><p>
- The library function which approximates a mathematical function shall signal
- a domain error whenever evaluated with argument values for which the mathematical
- function is undefined.
- </p></blockquote></div>
- <div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: disc; "><li class="listitem">
- Guideline 2
- </li></ul></div>
- <div class="blockquote"><blockquote class="blockquote"><p>
- The library function which approximates a mathematical function shall signal
- a domain error whenever evaluated with argument values for which the mathematical
- function obtains a non-real value.
- </p></blockquote></div>
- <p>
- This implementation is believed to follow these proposals and to assist compatibility
- with <span class="emphasis"><em>ISO/IEC 9899:1999 Programming languages - C</em></span> and with
- the <a href="http://www.open-std.org/jtc1/sc22/wg21/docs/papers/2005/n1836.pdf" target="_top">Draft
- Technical Report on C++ Library Extensions, 2005-06-24, section 5.2.1, paragraph
- 5</a>. <a class="link" href="error_handling.html" title="Error Handling">See also domain_error</a>.
- </p>
- <p>
- See <a class="link" href="pol_ref.html" title="Policy Reference">policy reference</a> for details
- of the error handling policies that should allow a user to comply with any
- of these recommendations, as well as other behaviour.
- </p>
- <p>
- See <a class="link" href="error_handling.html" title="Error Handling">error handling</a> for a
- detailed explanation of the mechanism, and <a class="link" href="stat_tut/weg/error_eg.html" title="Error Handling Example">error_handling
- example</a> and <a href="../../../example/error_handling_example.cpp" target="_top">error_handling_example.cpp</a>
- </p>
- <div class="caution"><table border="0" summary="Caution">
- <tr>
- <td rowspan="2" align="center" valign="top" width="25"><img alt="[Caution]" src="../../../../../doc/src/images/caution.png"></td>
- <th align="left">Caution</th>
- </tr>
- <tr><td align="left" valign="top"><p>
- If you enable throw but do NOT have try & catch block, then the program
- will terminate with an uncaught exception and probably abort. Therefore to
- get the benefit of helpful error messages, enabling <span class="bold"><strong>all</strong></span>
- exceptions <span class="bold"><strong>and</strong></span> using try&catch is recommended
- for all applications. However, for simplicity, this is not done for most
- examples.
- </p></td></tr>
- </table></div>
- <h5>
- <a name="math_toolkit.sf_implementation.h4"></a>
- <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
- of Functions that are Not Mathematically defined</a>
- </h5>
- <p>
- Functions that are not mathematically defined, like the Cauchy mean, fail to
- compile by default. A <a class="link" href="pol_ref/assert_undefined.html" title="Mathematically Undefined Function Policies">policy</a>
- allows control of this.
- </p>
- <p>
- If the policy is to permit undefined functions, then calling them throws a
- domain error, by default. But the error policy can be set to not throw, and
- to return NaN instead. For example,
- </p>
- <p>
- <code class="computeroutput"><span class="preprocessor">#define</span> <span class="identifier">BOOST_MATH_DOMAIN_ERROR_POLICY</span>
- <span class="identifier">ignore_error</span></code>
- </p>
- <p>
- appears before the first Boost include, then if the un-implemented function
- is called, mean(cauchy<>()) will return std::numeric_limits<T>::quiet_NaN().
- </p>
- <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>
- 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
- T is a User-defined type without NaN support), then an exception will always
- be thrown when a domain error occurs. Catching exceptions is therefore strongly
- recommended.
- </p></td></tr>
- </table></div>
- <h5>
- <a name="math_toolkit.sf_implementation.h5"></a>
- <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
- distributions</a>
- </h5>
- <p>
- There are many distributions for which we have been unable to find an analytic
- formula, and this has deterred us from implementing <a href="http://en.wikipedia.org/wiki/Median" target="_top">median
- functions</a>, the mid-point in a list of values.
- </p>
- <p>
- However a useful numerical approximation for distribution <code class="computeroutput"><span class="identifier">dist</span></code>
- 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
- an analytic formula) by calling
- </p>
- <p>
- <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>
- definition of course).
- </p>
- <p>
- <a href="http://www.amstat.org/publications/jse/v13n2/vonhippel.html" target="_top">Mean,
- Median, and Skew, Paul T von Hippel</a>
- </p>
- <p>
- <a href="http://documents.wolfram.co.jp/teachersedition/MathematicaBook/24.5.html" target="_top">Descriptive
- Statistics,</a>
- </p>
- <p>
- <a href="http://documents.wolfram.co.jp/v5/Add-onsLinks/StandardPackages/Statistics/DescriptiveStatistics.html" target="_top">and
- </a>
- </p>
- <p>
- <a href="http://documents.wolfram.com/v5/TheMathematicaBook/AdvancedMathematicsInMathematica/NumericalOperationsOnData/3.8.1.html" target="_top">Mathematica
- Basic Statistics.</a> give more detail, in particular for discrete distributions.
- </p>
- <h5>
- <a name="math_toolkit.sf_implementation.h6"></a>
- <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
- of Floating-Point Infinity</a>
- </h5>
- <p>
- Some functions and distributions are well defined with + or - infinity as argument(s),
- but after some experiments with handling infinite arguments as special cases,
- we concluded that it was generally more useful to forbid this, and instead
- to return the result of <a class="link" href="error_handling.html#math_toolkit.error_handling.domain_error">domain_error</a>.
- </p>
- <p>
- Handling infinity as special cases is additionally complicated because, unlike
- built-in types on most - but not all - platforms, not all User-Defined Types
- 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
- of infinity.
- </p>
- <p>
- The rationale is that non-finiteness may happen because of error or overflow
- in the users code, and it will be more helpful for this to be diagnosed promptly
- rather than just continuing. The code also became much more complicated, more
- error-prone, much more work to test, and much less readable.
- </p>
- <p>
- However in a few cases, for example normal, where we felt it obvious, we have
- permitted argument(s) to be infinity, provided infinity is implemented for
- the <code class="computeroutput"><span class="identifier">RealType</span></code> on that implementation,
- and it is supported and tested by the distribution.
- </p>
- <p>
- The range for these distributions is set to infinity if supported by the platform,
- (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
- for the <code class="computeroutput"><span class="identifier">RealType</span></code> by Boost.Math.
- </p>
- <p>
- Testing for has_infinity is obviously important for arbitrary precision types
- where infinity makes much less sense than for IEEE754 floating-point.
- </p>
- <p>
- 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
- the PDF is uninteresting/zero for infinities.
- </p>
- <p>
- Users who require special handling of infinity (or other specific value) can,
- of course, always intercept this before calling a distribution or function
- and return their own choice of value, or other behavior. This will often be
- simpler than trying to handle the aftermath of the error policy.
- </p>
- <p>
- Overflow, underflow, denorm can be handled using <a class="link" href="pol_ref/error_handling_policies.html" title="Error Handling Policies">error
- handling policies</a>.
- </p>
- <p>
- We have also tried to catch boundary cases where the mathematical specification
- would result in divide by zero or overflow and signalling these similarly.
- 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
- handling policies</a>.
- </p>
- <h5>
- <a name="math_toolkit.sf_implementation.h7"></a>
- <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
- and Location</a>
- </h5>
- <p>
- We considered adding location and scale to the list of functions, for example:
- </p>
- <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>
- <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>
- <span class="special">{</span>
- <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>
- <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>
- <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>
- <span class="identifier">RealType</span> <span class="identifier">result</span><span class="special">;</span> <span class="comment">// of checks.</span>
- <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>
- <span class="special">{</span>
- <span class="keyword">return</span> <span class="identifier">result</span><span class="special">;</span>
- <span class="special">}</span>
- <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>
- <span class="special">}</span>
- </pre>
- <p>
- but found that these concepts are not defined (or their definition too contentious)
- for too many distributions to be generally applicable. Because they are non-member
- functions, they can be added if required.
- </p>
- <h5>
- <a name="math_toolkit.sf_implementation.h8"></a>
- <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
- on Implementation of Specific Functions & Distributions</a>
- </h5>
- <div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: disc; "><li class="listitem">
- Default parameters for the Triangular Distribution. We are uncertain about
- the best default parameters. Some sources suggest that the Standard Triangular
- Distribution has lower = 0, mode = half and upper = 1. However as a approximation
- for the normal distribution, the most common usage, lower = -1, mode =
- 0 and upper = 1 would be more suitable.
- </li></ul></div>
- <h5>
- <a name="math_toolkit.sf_implementation.h9"></a>
- <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
- Approximations Used</a>
- </h5>
- <p>
- Some of the special functions in this library are implemented via rational
- approximations. These are either taken from the literature, or devised by John
- Maddock using <a class="link" href="internals/minimax.html" title="Minimax Approximations and the Remez Algorithm">our Remez code</a>.
- </p>
- <p>
- Rational rather than Polynomial approximations are used to ensure accuracy:
- polynomial approximations are often wonderful up to a certain level of accuracy,
- but then quite often fail to provide much greater accuracy no matter how many
- more terms are added.
- </p>
- <p>
- Our own approximations were devised either for added accuracy (to support 128-bit
- long doubles for example), or because literature methods were unavailable or
- under non-BSL compatible license. Our Remez code is known to produce good agreement
- with literature results in fairly simple "toy" cases. All approximations
- were checked for convergence and to ensure that they were not ill-conditioned
- (the coefficients can give a theoretically good solution, but the resulting
- rational function may be un-computable at fixed precision).
- </p>
- <p>
- Recomputing using different Remez implementations may well produce differing
- coefficients: the problem is well known to be ill conditioned in general, and
- our Remez implementation often found a broad and ill-defined minima for many
- of these approximations (of course for simple "toy" examples like
- approximating <code class="computeroutput"><span class="identifier">exp</span></code> the minima
- is well defined, and the coefficients should agree no matter whose Remez implementation
- is used). This should not in general effect the validity of the approximations:
- there's good literature supporting the idea that coefficients can be "in
- error" without necessarily adversely effecting the result. Note that "in
- error" has a special meaning in this context, see <a href="http://front.math.ucdavis.edu/0101.5042" target="_top">"Approximate
- construction of rational approximations and the effect of error autocorrection.",
- Grigori Litvinov, eprint arXiv:math/0101042</a>. Therefore the coefficients
- still need to be accurately calculated, even if they can be in error compared
- to the "true" minimax solution.
- </p>
- <h5>
- <a name="math_toolkit.sf_implementation.h10"></a>
- <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
- of Mathematical Constants</a>
- </h5>
- <p>
- A macro BOOST_DEFINE_MATH_CONSTANT in constants.hpp is used to provide high
- accuracy constants to mathematical functions and distributions, since it is
- important to provide values uniformly for both built-in float, double and long
- double types, and for User Defined types in <a href="../../../../../libs/multiprecision/doc/html/index.html" target="_top">Boost.Multiprecision</a>
- like <a href="../../../../../libs/multiprecision/doc/html/boost_multiprecision/tut/floats/cpp_dec_float.html" target="_top">cpp_dec_float</a>.
- and others like NTL::quad_float and NTL::RR.
- </p>
- <p>
- To permit calculations in this Math ToolKit and its tests, (and elsewhere)
- at about 100 decimal digits with NTL::RR type, it is obviously necessary to
- define constants to this accuracy.
- </p>
- <p>
- However, some compilers do not accept decimal digits strings as long as this.
- So the constant is split into two parts, with the 1st containing at least long
- double precision, and the 2nd zero if not needed or known. The 3rd part permits
- an exponent to be provided if necessary (use zero if none) - the other two
- parameters may only contain decimal digits (and sign and decimal point), and
- may NOT include an exponent like 1.234E99 (nor a trailing F or L). The second
- digit string is only used if T is a User-Defined Type, when the constant is
- converted to a long string literal and lexical_casted to type T. (This is necessary
- because you can't use a numeric constant since even a long double might not
- have enough digits).
- </p>
- <p>
- For example, pi is defined:
- </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">3.141592653589793238462643383279502884197169399375105820974944</span><span class="special">,</span>
- <span class="number">5923078164062862089986280348253421170679821480865132823066470938446095505</span><span class="special">,</span>
- <span class="number">0</span><span class="special">)</span>
- </pre>
- <p>
- And used thus:
- </p>
- <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>
- <span class="keyword">double</span> <span class="identifier">diameter</span> <span class="special">=</span> <span class="number">1.</span><span class="special">;</span>
- <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>
- <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>
- </pre>
- <p>
- Note that it is necessary (if inconvenient) to specify the type explicitly.
- </p>
- <p>
- So you cannot write
- </p>
- <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>
- </pre>
- <p>
- Neither can you write:
- </p>
- <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>
- <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>
- </pre>
- <h5>
- <a name="math_toolkit.sf_implementation.h11"></a>
- <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
- safety</a>
- </h5>
- <p>
- Reporting of error by setting <code class="computeroutput"><span class="identifier">errno</span></code>
- should be thread-safe already (otherwise none of the std lib math functions
- would be thread safe?). If you turn on reporting of errors via exceptions,
- <code class="computeroutput"><span class="identifier">errno</span></code> gets left unused anyway.
- </p>
- <p>
- For normal C++ usage, the Boost.Math <code class="computeroutput"><span class="keyword">static</span>
- <span class="keyword">const</span></code> constants are now thread-safe
- so for built-in real-number types: <code class="computeroutput"><span class="keyword">float</span></code>,
- <code class="computeroutput"><span class="keyword">double</span></code> and <code class="computeroutput"><span class="keyword">long</span>
- <span class="keyword">double</span></code> are all thread safe.
- </p>
- <p>
- 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>,
- the Boost.Math should also be thread-safe, (thought we are unsure how to rigorously
- prove this).
- </p>
- <p>
- (Thread safety has received attention in the C++11 Standard revision, so hopefully
- all compilers will do the right thing here at some point.)
- </p>
- <h5>
- <a name="math_toolkit.sf_implementation.h12"></a>
- <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
- of Test Data</a>
- </h5>
- <p>
- We found a large number of sources of test data. We have assumed that these
- are <span class="emphasis"><em>"known good"</em></span> if they agree with the results
- from our test and only consulted other sources for their <span class="emphasis"><em>'vote'</em></span>
- in the case of serious disagreement. The accuracy, actual and claimed, vary
- very widely. Only <a href="http://functions.wolfram.com/" target="_top">Wolfram Mathematica
- functions</a> provided a higher accuracy than C++ double (64-bit floating-point)
- and was regarded as the most-trusted source by far. The <a href="http://www.r-project.org/" target="_top">The
- R Project for Statistical Computing</a> provided the widest range of distributions,
- but the usual Intel X86 distribution uses 64-but doubles, so our use was limited
- to the 15 to 17 decimal digit accuracy.
- </p>
- <p>
- A useful index of sources is: <a href="http://www.sal.hut.fi/Teaching/Resources/ProbStat/table.html" target="_top">Web-oriented
- Teaching Resources in Probability and Statistics</a>
- </p>
- <p>
- <a href="http://espse.ed.psu.edu/edpsych/faculty/rhale/hale/507Mat/statlets/free/pdist.htm" target="_top">Statlet</a>:
- Is a Javascript application that calculates and plots probability distributions,
- and provides the most complete range of distributions:
- </p>
- <div class="blockquote"><blockquote class="blockquote"><p>
- Bernoulli, Binomial, discrete uniform, geometric, hypergeometric, negative
- binomial, Poisson, beta, Cauchy-Lorentz, chi-sequared, Erlang, exponential,
- extreme value, Fisher, gamma, Laplace, logistic, lognormal, normal, Parteo,
- Student's t, triangular, uniform, and Weibull.
- </p></blockquote></div>
- <p>
- It calculates pdf, cdf, survivor, log survivor, hazard, tail areas, & critical
- values for 5 tail values.
- </p>
- <p>
- It is also the only independent source found for the Weibull distribution;
- unfortunately it appears to suffer from very poor accuracy in areas where the
- underlying special function is known to be difficult to implement.
- </p>
- <h5>
- <a name="math_toolkit.sf_implementation.h13"></a>
- <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
- for Invalid Parameters to Functions and Constructors</a>
- </h5>
- <p>
- 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
- function
- </p>
- <p>
- <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
- BOOST_CHECK_THROW macro) that bad parameters passed to constructors and functions
- throw <code class="computeroutput"><span class="identifier">domain_error</span></code> exceptions.
- </p>
- <p>
- Usage is <code class="computeroutput"><span class="identifier">check_out_of_range</span><span class="special"><</span> <span class="identifier">DistributionType</span>
- <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>
- Where list-of-params is a list of <span class="bold"><strong>valid</strong></span> parameters
- from which the distribution can be constructed - ie the same number of args
- are passed to the function, as are passed to the distribution constructor.
- </p>
- <p>
- The values of the parameters are not important, but must be <span class="bold"><strong>valid</strong></span>
- to pass the constructor checks; the default values are suitable, but must be
- explicitly provided, for example:
- </p>
- <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>
- </pre>
- <p>
- Checks made are:
- </p>
- <div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: disc; ">
- <li class="listitem">
- Infinity or NaN (if available) passed in place of each of the valid params.
- </li>
- <li class="listitem">
- Infinity or NaN (if available) as a random variable.
- </li>
- <li class="listitem">
- Out-of-range random variable passed to pdf and cdf (ie outside of "range(DistributionType)").
- </li>
- <li class="listitem">
- Out-of-range probability passed to quantile function and complement.
- </li>
- </ul></div>
- <p>
- but does <span class="bold"><strong>not</strong></span> check finite but out-of-range
- parameters to the constructor because these are specific to each distribution,
- for example:
- </p>
- <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>
- <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>
- </pre>
- <p>
- 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
- that <code class="computeroutput"><span class="identifier">domain_error</span></code> exception
- is thrown if either are == 0.
- </p>
- <p>
- (Use of <code class="computeroutput"><span class="identifier">check_out_of_range</span></code>
- function may mean that some previous tests are now redundant).
- </p>
- <p>
- It was also noted that if more than one parameter is bad, then only the first
- detected will be reported by the error message.
- </p>
- <h5>
- <a name="math_toolkit.sf_implementation.h14"></a>
- <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
- and Managing the Equations</a>
- </h5>
- <p>
- Equations that fit on a single line can most easily be produced by inline Quickbook
- code using templates for Unicode Greek and Unicode Math symbols. All Greek
- letter and small set of Math symbols is available at /boost-path/libs/math/doc/sf_and_dist/html4_symbols.qbk
- </p>
- <p>
- Where equations need to use more than one line, real Math editors were used.
- </p>
- <p>
- The primary source for the equations is now <a href="http://www.w3.org/Math/" target="_top">MathML</a>:
- see the *.mml files in libs/math/doc/sf_and_dist/equations/.
- </p>
- <p>
- These are most easily edited by a GUI editor such as <a href="http://mathcast.sourceforge.net/home.html" target="_top">Mathcast</a>,
- please note that the equation editor supplied with Open Office currently mangles
- these files and should not currently be used.
- </p>
- <p>
- Conversion to SVG was achieved using <a href="https://sourceforge.net/projects/svgmath/" target="_top">SVGMath</a>
- and a command line such as:
- </p>
- <pre class="programlisting">$for file in *.mml; do
- >/cygdrive/c/Python25/python.exe 'C:\download\open\SVGMath-0.3.1\math2svg.py' \
- >>$file > $(basename $file .mml).svg
- >done
- </pre>
- <p>
- See also the section on "Using Python to run Inkscape" and "Using
- inkscape to convert scalable vector SVG files to Portable Network graphic PNG".
- </p>
- <p>
- Note that SVGMath requires that the mml files are <span class="bold"><strong>not</strong></span>
- wrapped in an XHTML XML wrapper - this is added by Mathcast by default - one
- workaround is to copy an existing mml file and then edit it with Mathcast:
- the existing format should then be preserved. This is a bug in the XML parser
- used by SVGMath which the author is aware of.
- </p>
- <p>
- If necessary the XHTML wrapper can be removed with:
- </p>
- <pre class="programlisting">cat filename | tr -d "\r\n" | sed -e 's/.*\(<math[^>]*>.*</math>\).*/\1/' > newfile</pre>
- <p>
- Setting up fonts for SVGMath is currently rather tricky, on a Windows XP system
- JM's font setup is the same as the sample config file provided with SVGMath
- but with:
- </p>
- <pre class="programlisting"> <!-- Double-struck -->
- <mathvariant name="double-struck" family="Mathematica7, Lucida Sans Unicode"/>
- </pre>
- <p>
- changed to:
- </p>
- <pre class="programlisting"> <!-- Double-struck -->
- <mathvariant name="double-struck" family="Lucida Sans Unicode"/>
- </pre>
- <p>
- Note that unlike the sample config file supplied with SVGMath, this does not
- make use of the <a href="http://support.wolfram.com/technotes/fonts/windows/latestfonts.html" target="_top">Mathematica
- 7 font</a> as this lacks sufficient Unicode information for it to be used
- with either SVGMath or XEP "as is".
- </p>
- <p>
- Also note that the SVG files in the repository are almost certainly Windows-specific
- since they reference various Windows Fonts.
- </p>
- <p>
- PNG files can be created from the SVGs using <a href="http://xmlgraphics.apache.org/batik/tools/rasterizer.html" target="_top">Batik</a>
- and a command such as:
- </p>
- <pre class="programlisting">java -jar 'C:\download\open\batik-1.7\batik-rasterizer.jar' -dpi 120 *.svg</pre>
- <p>
- Or using Inkscape (File, Export bitmap, Drawing tab, bitmap size (default size,
- 100 dpi), Filename (default). png)
- </p>
- <p>
- or Using Cygwin, a command such as:
- </p>
- <pre class="programlisting">for file in *.svg; do
- /cygdrive/c/progra~1/Inkscape/inkscape -d 120 -e $(cygpath -a -w $(basename $file .svg).png) $(cygpath -a -w $file);
- done</pre>
- <p>
- Using BASH
- </p>
- <pre class="programlisting"># Convert single SVG to PNG file.
- # /c/progra~1/Inkscape/inkscape -d 120 -e a.png a.svg
- </pre>
- <p>
- or to convert All files in folder SVG to PNG.
- </p>
- <pre class="programlisting">for file in *.svg; do
- /c/progra~1/Inkscape/inkscape -d 120 -e $(basename $file .svg).png $file
- done
- </pre>
- <p>
- Currently Inkscape seems to generate the better looking PNGs.
- </p>
- <p>
- The PDF is generated into \pdf\math.pdf using a command from a shell or command
- window with current directory \math_toolkit\libs\math\doc\sf_and_dist, typically:
- </p>
- <pre class="programlisting">bjam -a pdf >math_pdf.log</pre>
- <p>
- Note that XEP will have to be configured to <span class="bold"><strong>use and embed</strong></span>
- whatever fonts are used by the SVG equations (almost certainly editing the
- sample xep.xml provided by the XEP installation). If you fail to do this you
- will get XEP warnings in the log file like
- </p>
- <pre class="programlisting">[warning]could not find any font family matching "Times New Roman"; replaced by Helvetica</pre>
- <p>
- (html is the default so it is generated at libs\math\doc\html\index.html using
- command line >bjam -a > math_toolkit.docs.log).
- </p>
- <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>
- </pre>
- <p>
- is provided in the xep.xml downloaded, but the Windows TrueType fonts are commented
- out.
- </p>
- <p>
- JM's XEP config file \xep\xep.xml has the following font configuration section
- added:
- </p>
- <pre class="programlisting"> <font-group xml:base="file:/C:/Windows/Fonts/" label="Windows TrueType" embed="true" subset="true">
- <font-family name="Arial">
- <font><font-data ttf="arial.ttf"/></font>
- <font style="oblique"><font-data ttf="ariali.ttf"/></font>
- <font weight="bold"><font-data ttf="arialbd.ttf"/></font>
- <font weight="bold" style="oblique"><font-data ttf="arialbi.ttf"/></font>
- </font-family>
- <font-family name="Times New Roman" ligatures="&#xFB01; &#xFB02;">
- <font><font-data ttf="times.ttf"/></font>
- <font style="italic"><font-data ttf="timesi.ttf"/></font>
- <font weight="bold"><font-data ttf="timesbd.ttf"/></font>
- <font weight="bold" style="italic"><font-data ttf="timesbi.ttf"/></font>
- </font-family>
- <font-family name="Courier New">
- <font><font-data ttf="cour.ttf"/></font>
- <font style="oblique"><font-data ttf="couri.ttf"/></font>
- <font weight="bold"><font-data ttf="courbd.ttf"/></font>
- <font weight="bold" style="oblique"><font-data ttf="courbi.ttf"/></font>
- </font-family>
- <font-family name="Tahoma" embed="true">
- <font><font-data ttf="tahoma.ttf"/></font>
- <font weight="bold"><font-data ttf="tahomabd.ttf"/></font>
- </font-family>
- <font-family name="Verdana" embed="true">
- <font><font-data ttf="verdana.ttf"/></font>
- <font style="oblique"><font-data ttf="verdanai.ttf"/></font>
- <font weight="bold"><font-data ttf="verdanab.ttf"/></font>
- <font weight="bold" style="oblique"><font-data ttf="verdanaz.ttf"/></font>
- </font-family>
- <font-family name="Palatino" embed="true" ligatures="&#xFB00; &#xFB01; &#xFB02; &#xFB03; &#xFB04;">
- <font><font-data ttf="pala.ttf"/></font>
- <font style="italic"><font-data ttf="palai.ttf"/></font>
- <font weight="bold"><font-data ttf="palab.ttf"/></font>
- <font weight="bold" style="italic"><font-data ttf="palabi.ttf"/></font>
- </font-family>
- <font-family name="Lucida Sans Unicode">
- <!-- <font><font-data ttf="lsansuni.ttf"><<span class="emphasis"><em>font> -->
- <!-- actually called l_10646.ttf on Windows 2000 and Vista Sp1 -->
- <font><font-data ttf="l_10646.ttf"</em></span>></font>
- </font-family>
- </pre>
- <p>
- PAB had to alter his because the Lucida Sans Unicode font had a different name.
- Other changes are very likely to be required if you are not using Windows.
- </p>
- <p>
- XZ authored his equations using the venerable Latex, JM converted these to
- MathML using <a href="http://gentoo-wiki.com/HOWTO_Convert_LaTeX_to_HTML_with_MathML" target="_top">mxlatex</a>.
- This process is currently unreliable and required some manual intervention:
- consequently Latex source is not considered a viable route for the automatic
- production of SVG versions of equations.
- </p>
- <p>
- Equations are embedded in the quickbook source using the <span class="emphasis"><em>equation</em></span>
- template defined in math.qbk. This outputs Docbook XML that looks like:
- </p>
- <pre class="programlisting"><inlinemediaobject>
- <imageobject role="html">
- <imagedata fileref="../equations/myfile.png"></imagedata>
- </imageobject>
- <imageobject role="print">
- <imagedata fileref="../equations/myfile.svg"></imagedata>
- </imageobject>
- </inlinemediaobject>
- </pre>
- <p>
- MathML is not currently present in the Docbook output, or in the generated
- HTML: this needs further investigation.
- </p>
- <h5>
- <a name="math_toolkit.sf_implementation.h15"></a>
- <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
- Graphs</a>
- </h5>
- <p>
- Graphs were produced in SVG format and then converted to PNG's using the same
- process as the equations.
- </p>
- <p>
- 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
- the SVG's directly using the <a href="http://code.google.com/soc/2007/boost/about.html" target="_top">Google
- Summer of Code 2007</a> project of Jacob Voytko (whose work so far, considerably
- enhanced and now reasonably mature and usable, by Paul A. Bristow, is at .\boost-sandbox\SOC\2007\visualization).
- </p>
- </div>
- <table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr>
- <td align="left"></td>
- <td align="right"><div class="copyright-footer">Copyright © 2006-2019 Nikhar
- Agrawal, Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos,
- Hubert Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Matthew Pulver, Johan
- Råde, Gautam Sewani, Benjamin Sobotta, Nicholas Thompson, 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>)
- </p>
- </div></td>
- </tr></table>
- <hr>
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