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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.

##### Implemention philosophy

"First be right, then be fast."

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.

So our priority is accuracy.

To permit evaluation of accuracy of the special functions, production of extremely accurate tables of test values has received considerable effort.

(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 days calculating some tables of test values!)

For a specific RealType, say float or double, 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).

If these prove accurate enough to be "fit for his purpose", then a user may substitute his custom specialization.

For example, there are approximations dating back from times when computation was a lot more expensive:

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).

A H Carter, Approximations to percentage points of the z-distribution, Biometrika 34(2), 352 - 358 (Dec 1947).

These could still provide sufficient accuracy for some speed-critical applications.

##### Accuracy and Representation of Test Values

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 static_cast<RealType>(1.2345L) to provide the appropriate RealType for spot tests.

Functions that return constants values, like kurtosis for example, are written as

static_cast<RealType>(-3) / 5;

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).

So tests for one third, not exactly representable with radix two floating-point, (should) use, for example:

static_cast<RealType>(1) / 3;

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 a/b with b a power of two (in order to ensure that the result is an exactly representable binary value).

##### Tolerance of Tests

The tolerances need to be set to the maximum of:

• Some epsilon value.
• The accuracy of the data (often only near 64-bit double).

Otherwise when long double has more digits than the test data, then no amount of tweaking an epsilon based tolerance will work.

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.

##### Handling Unsuitable Arguments

In Errors in Mathematical Special Functions, J. Marraffino & M. Paterno it is proposed that signalling a domain error is mandatory when the argument would give an mathematically undefined result.

• Guideline 1

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.

Put crudely, if the function goes to + infinity and then emerges 'round-the-back' with - infinity, it is NOT defined.

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.

• Guideline 2

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.

This implementation is believed to follow these proposals and to assist compatibility with ISO/IEC 9899:1999 Programming languages - C and with the Draft Technical Report on C++ Library Extensions, 2005-06-24, section 5.2.1, paragraph 5. See also domain_error.

See policy reference for details of the error handling policies that should allow a user to comply with any of these recommendations, as well as other behaviour.

See error handling for a detailed explanation of the mechanism, and error_handling example and error_handling_example.cpp

Caution 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 all exceptions and using try&catch is recommended for all applications. However, for simplicity, this is not done for most examples.
##### Handling of Functions that are Not Mathematically defined

Functions that are not mathematically defined, like the Cauchy mean, fail to compile by default. A policy allows control of this.

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,

#define BOOST_MATH_DOMAIN_ERROR_POLICY ignore_error

appears before the first Boost include, then if the un-implemented function is called, mean(cauchy<>()) will return std::numeric_limits<T>::quiet_NaN().

Warning If std::numeric_limits::has_quiet_NaN is false (for example T is a User-defined type), then an exception will always be thrown when a domain error occurs. Catching exceptions is therefore strongly recommended.
##### Median of distributions

There are many distributions for which we have been unable to find an analytic formula, and this has deterred us from implementing median functions, the mid-point in a list of values.

However a useful median approximation for distribution dist may be available from

quantile(dist, 0.5).

Mathematica Basic Statistics. give more detail, in particular for discrete distributions.

##### Handling of Floating-Point Infinity

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 domain_error.

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 std::numeric_limits<RealType>::infinity() and would return zero rather than any representation of infinity.

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.

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 realType on that implementation.

Overflow, underflow, denorm can be handled using error handling policies.

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 error handling policies.

##### Scale, Shape and Location

We considered adding location and scale to the list of functions, for example:

template <class RealType>
inline RealType scale(const triangular_distribution<RealType>& dist)
{
RealType lower = dist.lower();
RealType mode = dist.mode();
RealType upper = dist.upper();
RealType result;  // of checks.
if(false == detail::check_triangular(BOOST_CURRENT_FUNCTION, lower, mode, upper, &result))
{
return result;
}
return (upper - lower);
}

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.

##### Notes on Implementation of Specific Functions & Distributions
• 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.
##### Rational Approximations Used

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 our Remez code.

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.

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).

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 exp the minima is well defined, and the coeffiecents 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 "Approximate construction of rational approximations and the effect of error autocorrection.", Grigori Litvinov, eprint arXiv:math/0101042. Therefore the coefficients still need to be accurately calculated, even if they can be in error compared to the "true" minimax solution.

##### Representation of Mathematical Constants

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 like NTL::quad_float and NTL::RR.

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.

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).

For example, pi is defined:

BOOST_DEFINE_MATH_CONSTANT(pi,
3.141592653589793238462643383279502884197169399375105820974944,
5923078164062862089986280348253421170679821480865132823066470938446095505,
0)

And used thus:

using namespace boost::math::constants;

double diameter = 1.;
double radius = diameter * pi<double>();

or boost::math::constants::pi<NTL::RR>()

Note that it is necessary (if inconvenient) to specify the type explicitly.

So you cannot write

double p = boost::math::constants::pi<>();  // could not deduce template argument for 'T'

Neither can you write:

double p = boost::math::constants::pi; // Context does not allow for disambiguation of overloaded function
double p = boost::math::constants::pi(); // Context does not allow for disambiguation of overloaded function

Reporting of error by setting errno 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, errno gets left unused anyway.

Other than that, the code is intended to be thread safe for built in real-number types : so float, double and long double are all thread safe.

For non-built-in types - NTL::RR for example - initialisation of the various constants used in the implementation is potentially not thread safe. This most undesiable, but it would be a signficant challenge to fix it. Some compilers may offer the option of having static-constants initialised in a thread safe manner (Commeau, and maybe others?), if that's the case then the problem is solved. This is a topic of hot debate for the next C++ std revision, so hopefully all compilers will be required to do the right thing here at some point.

##### Sources of Test Data

We found a large number of sources of test data. We have assumed that these are "known good" if they agree with the results from our test and only consulted other sources for their 'vote' in the case of serious disagreement. The accuracy, actual and claimed, vary very widely. Only Wolfram Mathematica functions provided a higher accuracy than C++ double (64-bit floating-point) and was regarded as the most-trusted source by far.

A useful index of sources is: Web-oriented Teaching Resources in Probability and Statistics

Statlet: Is a Javascript application that calculates and plots probability distributions, and provides the most complete range of distributions:

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.

It calculates pdf, cdf, survivor, log survivor, hazard, tail areas, & critical values for 5 tail values.

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.

##### Creating and Managing the Equations

The primary source for the equations is now MathML: see the *.mml files in libs/math/doc/sf_and_dist/equations/.

These are most easily edited by a GUI editor such as Mathcast, please note that the equation editor supplied with Open Office currently mangles these files and should not currently be used.

Convertion to SVG was achieved using SVGMath and a command line such as:

\$for file in *.mml; do
>>\$file > \$(basename \$file .mml).svg
>done

Note that SVGMath requires that the mml files are not 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.

If neccessary the XHTML wrapper can be removed with:

cat filename | tr -d "\r\n" | sed -e 's/.*\(<math[^>]*>.*[/itex]\).*/\1/' > newfile

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:

<!-- Double-struck -->
<mathvariant name="double-struck" family="Mathematica7, Lucida Sans Unicode"/>

changed to:

<!-- Double-struck -->
<mathvariant name="double-struck" family="Lucida Sans Unicode"/>

Note that unlike the sample config file supplied with SVGMath, this does not make use of the Mathematica 7 font as this lacks sufficient Unicode information for it to be used with either SVGMath or XEP "as is".

Also note that the SVG files in the repository are almost certainly Windows-specific since they reference various Windows Fonts.

PNG files can be created from the SVG's using Batik and a command such as:

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:

bjam -a pdf

Note that XEP will have to be configured to use and embed whatever fonts are used by the SVG equations (if necessary editing the sample xep.xml provided by the XEP installation).

(html is generated at math_toolkit\libs\math\doc\sf_and_dist\html\index.html using just bjam -a).

JM's XEP config file has the following font configuration section added:

<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"/></font>
</font-family>

PAB had to alter his because the Lucida Sans Unicode font had a different name. Changes are very likely to be required if you are not using Windows.

XZ authored his equations using the venerable Latex, JM converted these to MathML using mxlatex. 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.

Equations are embedded in the quickbook source using the equation template defined in math.qbk. This outputs Docbook XML that looks like:

<inlinemediaobject>
<imageobject role"html">
<imagedata fileref"../equations/myfile.png"></imagedata>
</imageobject>
<imageobject role"print">
<imagedata fileref"../equations/myfile.svg"></imagedata>
</imageobject>
</inlinemediaobject>

MathML is not currently present in the Docbook output, or in the generated HTML: this needs further investigation.

##### Producing Graphs

Graphs were mostly produced by a very laborious process entailing output of columns of values from C++ programs to a .csv file, use of RJS Graph to arrange the display and axes, and output to a .ps file, followed by conversion to .png using Adobe Photoshop, or similar utility. This rigmarole is not recommended!

We plan to carry out this process in a single step using the Google Summer of Code 2007 project of Jacob Voytko (whose work so far is at .\boost-sandbox\SOC\2007\visualization) that should, when completed, allow output of annotated graphs as Scalable Vector Graphic .svg files directly from C++ programs.