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The functions and statistical distributions in this library can be used with
any type RealType that meets the conceptual requirements
given below. All the built-in floating-point types like double
will meet these requirements. (Built-in types are also called fundamental
types).
User-defined types that meet the conceptual requirements can also be used. For example, with a thin wrapper class one of the types provided with NTL (RR) can be used. But now that Boost.Multiprecision library is available, this has become the preferred real-number type, typically cpp_dec_float or cpp_bin_float.
Submissions of binding to other extended precision types would also still be welcome.
The guiding principal behind these requirements is that a RealType behaves just like a built-in floating-point type.
These requirements are common to all of the functions in this library.
In the following table r is an object of type RealType
, cr and cr2
are objects of type const RealType
,
and ca is an object of type const
arithmetic-type
(arithmetic types include all the built
in integers and floating point types).
Expression |
Result Type |
Notes |
---|---|---|
|
RealType |
RealType is copy constructible. |
|
RealType |
RealType is copy constructible from the arithmetic types. |
|
RealType& |
Assignment operator. |
|
RealType& |
Assignment operator from the arithmetic types. |
|
RealType& |
Adds cr to r. |
|
RealType& |
Adds ar to r. |
|
RealType& |
Subtracts cr from r. |
|
RealType& |
Subtracts ca from r. |
|
RealType& |
Multiplies r by cr. |
|
RealType& |
Multiplies r by ca. |
|
RealType& |
Divides r by cr. |
|
RealType& |
Divides r by ca. |
|
RealType |
Unary Negation. |
|
RealType& |
Identity Operation. |
|
RealType |
Binary Addition |
|
RealType |
Binary Addition |
|
RealType |
Binary Addition |
|
RealType |
Binary Subtraction |
|
RealType |
Binary Subtraction |
|
RealType |
Binary Subtraction |
|
RealType |
Binary Multiplication |
|
RealType |
Binary Multiplication |
|
RealType |
Binary Multiplication |
|
RealType |
Binary Subtraction |
|
RealType |
Binary Subtraction |
|
RealType |
Binary Subtraction |
|
bool |
Equality Comparison |
|
bool |
Equality Comparison |
|
bool |
Equality Comparison |
|
bool |
Inequality Comparison |
|
bool |
Inequality Comparison |
|
bool |
Inequality Comparison |
|
bool |
Less than equal to. |
|
bool |
Less than equal to. |
|
bool |
Less than equal to. |
|
bool |
Greater than equal to. |
|
bool |
Greater than equal to. |
|
bool |
Greater than equal to. |
|
bool |
Less than comparison. |
|
bool |
Less than comparison. |
|
bool |
Less than comparison. |
|
bool |
Greater than comparison. |
|
bool |
Greater than comparison. |
|
bool |
Greater than comparison. |
|
int |
The number of digits in the significand of RealType. |
|
RealType |
The largest representable number by type RealType. |
|
RealType |
The smallest representable number by type RealType. |
|
RealType |
The natural logarithm of the largest representable number by type RealType. |
|
RealType |
The natural logarithm of the smallest representable number by type RealType. |
|
RealType |
The machine epsilon of RealType. |
Note that:
log_max_value
and log_min_value
can be
synthesised from the others, and so no explicit specialisation is required.
epsilon
can
be synthesised from the others, so no explicit specialisation is required
provided the precision of RealType does not vary at runtime (see the header
boost/math/bindings/rr.hpp
for an example where the precision does vary at runtime).
digits
,
max_value
and min_value
, all get synthesised automatically
from std::numeric_limits
. However, if numeric_limits
is not specialised for
type RealType, then you will get a compiler error when code tries to use
these functions, unless you explicitly specialise
them. For example if the precision of RealType varies at runtime, then
numeric_limits
support
may not be appropriate, see boost/math/bindings/rr.hpp
for examples.
Warning | |
---|---|
If Boost.Test: giving misleading error messages like "difference between {9.79796} and {9.79796} exceeds 5.42101e-19%". Boost.LexicalCast and Boost.Serialization when converting the number to a string, causing potentially serious loss of accuracy on output.
Although it might seem obvious that RealType should require |
Many (though not all) of the functions in this library make calls to standard library functions, the following table summarises the requirements. Note that most of the functions in this library will only call a small subset of the functions listed here, so if in doubt whether a user-defined type has enough standard library support to be useable the best advise is to try it and see!
In the following table r is an object of type RealType
, cr1 and cr2
are objects of type const RealType
,
and i is an object of type int
.
Expression |
Result Type |
---|---|
|
RealType |
|
RealType |
|
RealType |
|
RealType |
|
RealType |
|
RealType |
|
RealType |
|
RealType |
|
RealType |
|
RealType |
|
RealType |
|
RealType |
|
RealType |
|
RealType |
|
RealType |
|
RealType |
|
RealType |
|
int |
|
RealType |
|
int |
Note that the table above lists only those standard library functions known
to be used (or likely to be used in the near future) by this library. The following
functions: acos
, atan2
, fmod
,
cosh
, sinh
,
tanh
, log10
,
lround
, llround
,
ltrunc
, lltrunc
and modf
are not currently
used, but may be if further special functions are added.
Note that the round
, trunc
and modf
functions are not part of the current C++ standard: they are part of the additions
added to C99 which will likely be in the next C++ standard. There are Boost
versions of these provided as a backup, and the functions are always called
unqualified so that argument-dependent-lookup can take place.
In addition, for efficient and accurate results, a Lanczos
approximation is highly desirable. You may be able to adapt an existing
approximation from boost/math/special_functions/lanczos.hpp
or boost/math/bindings/detail/big_lanczos.hpp:
in the former case you will need change static_cast
's
to lexical_cast
's, and the
constants to strings (in order to ensure the coefficients
aren't truncated to long doubl
e)
and then specialise lanczos_traits
for type T. Otherwise you may have to hack libs/math/tools/lanczos_generator.cpp
to find a suitable approximation for your RealType. The code will still compile
if you don't do this, but both accuracy and efficiency will be greatly compromised
in any function that makes use of the gamma/beta/erf family of functions.