math/doc/implementation.qbk

[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 to 64-bit doubles).
Values are specified as long double types by appending L,
unless they are exactly representable, for example integers.
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 real type 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 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.

See [link math_toolkit.special.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 Implemented]

Functions that are not implemented for any reason,
usually because they are not defined, (or we were unable to find a definition),
are handled as domain errors.

If the instruction 

`#define BOOST_MATH_THROW_ON_DOMAIN_ERROR`

appears before the first Boost include, then if the un-implemented function is called,
a domain_error exception will be thrown,
and caught to provide an appropriate error message.

Otherwise the value of std::numeric_limits<T>::quiet_NaN() will be returned.

[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 median approximation for distribution `dist` may be available from

`quantile(dist, 0.5)`.

[@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 more useful to forbid this,
and instead to return the result of __domain_error.

Handling infinity as special cases was 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.

We have also tried to catch boundary cases where the mathematical specification
would result in divide by zero or overflow and signalling these similarly.

[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 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 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.

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.

[endsect][/section:implementation Implementation Notes]

[/ 
  Copyright 2006 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).
]