[section:implementation Additional Implementation Notes]

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.

[h4 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.

[h4 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).

[h4 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.

[h4 Handling Unsuitable Arguments]

In
[@http://www.open-std.org/jtc1/sc22/wg21/docs/papers/2004/n1665.pdf 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
[@http://www.open-std.org/jtc1/sc22/wg21/docs/papers/2005/n1836.pdf Draft Technical Report on C++ Library Extensions, 2005-06-24, section 5.2.1, paragraph 5].
[link math_toolkit.main_overview.error_handling See also domain_error].

See __policy_ref 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 [link math_toolkit.main_overview.error_handling error handling]
for a detailed explanation of the mechanism, and
[link math_toolkit.dist.stat_tut.weg.error_eg error_handling example]
and
[@../../../example/error_handling_example.cpp 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.]

[h4 Handling of Functions that are Not Mathematically defined]

Functions that are not mathematically defined,
like the Cauchy mean, fail to compile by default.
A [link math_toolkit.policy.pol_ref.assert_undefined 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<T>::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.]

[h4 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 
[@http://en.wikipedia.org/wiki/Median median functions], the mid-point in a list of values.

However a useful numerical approximation for distribution `dist`
is available as usual as an accessor non-member function median using `median(dist)`,
that may be evaluated (in the absence of an analytic formula) by calling

`quantile(dist, 0.5)` (this is the /mathematical/ definition of course).

[@http://www.amstat.org/publications/jse/v13n2/vonhippel.html Mean, Median, and Skew, Paul T von Hippel]

[@http://documents.wolfram.co.jp/teachersedition/MathematicaBook/24.5.html Descriptive Statistics,]

[@http://documents.wolfram.co.jp/v5/Add-onsLinks/StandardPackages/Statistics/DescriptiveStatistics.html and ]

[@http://documents.wolfram.com/v5/TheMathematicaBook/AdvancedMathematicsInMathematica/NumericalOperationsOnData/3.8.1.html
Mathematica Basic Statistics.] give more detail, in particular for discrete distributions.


[h4 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.

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.

Overflow, underflow, denorm can be handled using __error_policy.

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_policy.

[h4 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.

[h4 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.

[h4 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
[link math_toolkit.toolkit.internals2.minimax 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 [@http://front.math.ucdavis.edu/0101.5042
"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.

[h4 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

[h4 Thread safety]

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.

[h4 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 [@http://functions.wolfram.com/ Wolfram Mathematica functions]
provided a higher accuracy than
C++ double (64-bit floating-point) and was regarded as
the most-trusted source by far.
The __R 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.

A useful index of sources is:
[@http://www.sal.hut.fi/Teaching/Resources/ProbStat/table.html
Web-oriented Teaching Resources in Probability and Statistics]

[@http://espse.ed.psu.edu/edpsych/faculty/rhale/hale/507Mat/statlets/free/pdist.htm 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.

[h4 Creating and Managing the Equations]

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

Where equations need to use more than one line, real Math editors were used.

The primary source for the equations is now
[@http://www.w3.org/Math/ MathML]: see the
*.mml files in libs\/math\/doc\/sf_and_dist\/equations\/.

These are most easily edited by a GUI editor such as
[@http://mathcast.sourceforge.net/home.html Mathcast],
please note that the equation editor supplied with Open Office
currently mangles these files and should not currently be used.

Conversion to SVG was achieved using
[@http://www.grigoriev.ru/svgmath/ SVGMath] and a command line
such as:

[pre
$for file in *.mml; do
>/cygdrive/c/Python25/python.exe 'C:\download\open\SVGMath-0.3.1\math2svg.py' \\
>>$file > $(basename $file .mml).svg
>done
]

See also the section on "Using Python to run Inkscape" and
"Using inkscape to convert scalable vector SVG files to Portable Network graphic PNG".

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:

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

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

changed to:

[pre
    <!\-\- 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
[@http://xmlgraphics.apache.org/batik/tools/rasterizer.html Batik]
and a command such as:

[pre java -jar 'C:\download\open\batik-1.7\batik-rasterizer.jar' -dpi 120 *.svg]

Or using Inkscape and a command such as:

[pre 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]

Currently Inkscape seems to generate the better looking png's.

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:

[pre bjam -a pdf >math_pdf.log]

Note that XEP will have to be configured to *use and embed*
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

[pre \[warning\]could not find any font family matching "Times New Roman"; replaced by Helvetica]

(html is the default so it is generated at math_toolkit\libs\math\doc\sf_and_dist\html\index.html
using command line >bjam -a > math_docs.log).

 <!-- Sample configuration for Windows TrueType fonts.  -->
is provided in the xep.xml downloaded, but the Windows TrueType fonts are commented out.

JM's XEP config file \xep\xep.xml has the following font configuration section added:

[pre
    <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> -->
         <!-- actually called l_10646.ttf on Windows 2000 and Vista Sp1 -->
         <font><font-data ttf="l_10646.ttf"/></font>
    </font-family>
]

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.

XZ authored his equations using the venerable Latex, JM converted these to
MathML using [@http://gentoo-wiki.com/HOWTO_Convert_LaTeX_to_HTML_with_MathML 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:

[pre
<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.

[h4 Producing Graphs]

Graphs were produced in SVG format and then converted to PNG's using the same
process as the equations.  

The programs 
`/libs/math/doc/sf_and_dist/graphs/dist_graphs.cpp`
and `/libs/math/doc/sf_and_dist/graphs/sf_graphs.cpp`
generate the SVG's directly using the 
[@http://code.google.com/soc/2007/boost/about.html Google Summer of Code 2007]
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).

[endsect] [/section:implementation Implementation Notes]

[/
  Copyright 2006, 2007, 2010 John Maddock and Paul A. Bristow.
  Distributed under the Boost Software License, Version 1.0.
  (See accompanying file LICENSE_1_0.txt or copy at
  http://www.boost.org/LICENSE_1_0.txt).
]