...one of the most highly
regarded and expertly designed C++ library projects in the
world.

— Herb Sutter and Andrei
Alexandrescu, C++
Coding Standards

As implied by its name, this library is intended to help manipulating
mathematical intervals. It consists of a single header <boost/numeric/interval.hpp>
and principally a type which can be used as `interval<T>`

.
In fact, this interval template is declared as
`interval<T,Policies>`

where `Policies`

is a
policy class that controls the various behaviours of the interval class;
`interval<T>`

just happens to pick the default policies for
the type `T`

.

An interval is a pair of numbers which represents all the numbers between
these two. (Intervals are considered close so the bounds are included.) The
purpose of this library is to extend the usual arithmetic functions to
intervals. These intervals will be written [*a*,*b*] to represent
all the numbers between *a* and *b* (included). *a* and
*b* can be infinite (but they can not be the same infinite) and *a*
≤ *b*.

The fundamental property of interval arithmetic is the
* inclusion property*:

- ``if
*f*is a function on a set of numbers,*f*can be extended to a new function defined on intervals. This new function*f*takes one interval argument and returns an interval result such as: ∀*x*∈ [*a*,*b*],*f*(*x*) ∈*f*([*a*,*b*]).''

Such a property is not limited to functions with only one argument. Whenever possible, the interval result should be the smallest one able to satisfy the property (it is not really useful if the new functions always answer [-∞,+∞]).

There are at least two reasons a user would like to use this library. The obvious one is when the user has to compute with intervals. One example is when input data have some builtin imprecision: instead of a number, an input variable can be passed as an interval. Another example application is to solve equations, by bisecting an interval until the interval width is small enough. A third example application is in computer graphics, where computations with boxes, segments or rays can be reduced to computations with points via intervals.

Another common reason to use interval arithmetic is when the computer doesn't produce exact results: by using intervals, it is possible to quantify the propagation of rounding errors. This approach is used often in numerical computation. For example, let's assume the computer stores numbers with ten decimal significant digits. To the question 1 + 1E-100 - 1, the computer will answer 0 although the correct answer would be 1E-100. With the help of interval arithmetic, the computer will answer [0,1E-9]. This is quite a huge interval for such a little result, but the precision is now known, without having to compute error propagation.

The * base number type* is the type that holds the
bounds of the interval. In order to successfully use interval arithmetic, the
base number type must present some characteristics. Firstly, due to the definition of an
interval, the base numbers have to be totally ordered so, for instance,

`complex<T>`

is not usable as base number type for
intervals. The mathematical functions for the base number type should also
be compatible with the total order (for instance if x>y and z>t, then
it should also hold that x+z > y+t), so modulo types are not usable
either.Secondly, the computations must be exact or provide some rounding methods (for instance, toward minus or plus infinity) if we want to guarantee the inclusion property. Note that we also may explicitely specify no rounding, for instance if the base number type is exact, i.e. the result of a mathematic operation is always computed and represented without loss of precision. If the number type is not exact, we may still explicitely specify no rounding, with the obvious consequence that the inclusion property is no longuer guaranteed.

Finally, because heavy loss of precision is always possible, some numbers
have to represent infinities or an exceptional behavior must be defined. The
same situation also occurs for NaN (*Not a Number*).

Given all this, one may want to limit the template argument T of the class
template `interval`

to the floating point types
`float`

, `double`

, and `long double`

, as
defined by the IEEE-754 Standard. Indeed, if the interval arithmetic is
intended to replace the arithmetic provided by the floating point unit of a
processor, these types are the best choice. Unlike `std::complex`

,
however, we don't want to limit `T`

to these types. This is why we
allow the rounding and exceptional behaviors to be given by the two policies
(rounding and checking). We do nevertheless provide highly optimized rounding
and checking class specializations for the above-mentioned floating point
types.

It is straightforward to define the elementary arithmetic operations on intervals, being guided by the inclusion property. For instance, if [a,b] and [c,d] are intervals, [a,b]+[c,d] can be computed by taking the smallest interval that contains all the numbers x+y for x in [a,b] and y in [c,d]; in this case, rounding a+b down and c+d up will suffice. Other operators and functions are similarly defined (see their definitions below).

It is also possible to define some comparison operators. Given two
intervals, the result is a tri-state boolean type
{*false*,*true,indeterminate*}. The answers *false* and
*true* are easy to manipulate since they can directly be mapped on the
boolean *true* and *false*. But it is not the case for the answer
*indeterminate* since comparison operators are supposed to be boolean
functions. So, what to do in order to obtain boolean answers?

One solution consists of deciding to adopt an exceptional behavior, such as a failed assertion or raising an exception. In this case, the exceptional behavior will be triggered when the result is indeterminate.

Another solution is to map *indeterminate* always to *false,*
or always to *true*. If *false* is chosen, the comparison will be
called "*certain*;" indeed, the result of [*a*,*b*] <
[*c*,*d*] will be *true* if and only if: ∀ *x*
∈ [*a*,*b*] ∀ *y* ∈ [*c*,*d*],
*x* < *y*. If *true* is chosen, the comparison will be
called "*possible*;" indeed, the result of [*a*,*b*] <
[*c*,*d*] will be *true* if and only if: ∃ *x*
∈ [*a*,*b*] ∃ *y* ∈ [*c*,*d*],
*x* < *y*.

Since any of these solution has a clearly defined semantics, it is not clear that we should enforce either of them. For this reason, the default behavior consists to mimic the real comparisons by throwing an exception in the indeterminate case. Other behaviors can be selected bu using specific comparison namespace. There is also a bunch of explicitely named comparison functions. See comparisons pages for further details.

This library provides two quite distinct levels of usage. One is to use
the basic class template `interval<T>`

without specifying
the policy. This only requires to know and understand the concepts developed
above and the content of the namespace boost. In addition to the class
`interval<T>`

, this level of usage provides arithmetic
operators (`+`

, `-`

, `*`

, `/`

),
algebraic and piecewise-algebraic functions (`abs`

,
`square`

, `sqrt`

, `pow`

), transcendental and
trigonometric functions (`exp`

, `log`

,
`sin`

, `cos`

, `tan`

, `asin`

,
`acos`

, `atan`

, `sinh`

, `cosh`

,
`tanh`

, `asinh`

, `acosh`

,
`atanh`

), and the standard comparison operators
(`<`

, `<=`

, `>`

,
`>=`

, `==`

, `!=`

), as well as several
interval-specific functions (`min`

, `max`

, which have a
different meaning than `std::min`

and `std::max`

;
`lower`

, `upper`

, `width`

,
`median`

, `empty`

, `singleton, equal`

,
`in`

, `in_zero`

, `subset`

,
`proper_subset`

, `overlap`

, `intersection`

,
`hull`

, `bisect`

).

For some functions which take several parameters of type
`interval<T>`

, all combinations of argument types
`T`

and `interval<T>`

which contain at least one
`interval<T>`

, are considered in order to avoid a conversion
from the arguments of type `T`

to a singleton of type
`interval<T>`

. This is done for efficiency reasons (the fact
that an argument is a singleton sometimes renders some tests unnecessary).

A somewhat more advanced usage of this library is to hand-pick the
policies `Rounding`

and `Checking`

and pass them to
`interval<T, Policies>`

through the use of ```
Policies :=
boost::interval_lib::policies<Rounding,Checking>
```

. Appropriate
policies can be fabricated by using the various classes provided in the
namespace `boost::interval_lib`

as detailed in section Interval Support Library. It is also possible to
choose the comparison scheme by overloading operators through namespaces.

namespace boost { namespace numeric { namespace interval_lib { /* this declaration is necessary for the declaration of interval */ template <class T> struct default_policies; /* ... ; the full synopsis of namespace interval_lib can be found here */ } // namespace interval_lib /* template interval_policies; class definition can be found here */ template<class Rounding, class Checking> struct interval_policies; /* template class interval; class definition can be found here */ template<class T, class Policies = typename interval_lib::default_policies<T>::type > class interval; /* arithmetic operators involving intervals */ template <class T, class Policies> interval<T, Policies> operator+(const interval<T, Policies>& x); template <class T, class Policies> interval<T, Policies> operator-(const interval<T, Policies>& x); template <class T, class Policies> interval<T, Policies> operator+(const interval<T, Policies>& x, const interval<T, Policies>& y); template <class T, class Policies> interval<T, Policies> operator+(const interval<T, Policies>& x, const T& y); template <class T, class Policies> interval<T, Policies> operator+(const T& x, const interval<T, Policies>& y); template <class T, class Policies> interval<T, Policies> operator-(const interval<T, Policies>& x, const interval<T, Policies>& y); template <class T, class Policies> interval<T, Policies> operator-(const interval<T, Policies>& x, const T& y); template <class T, class Policies> interval<T, Policies> operator-(const T& x, const interval<T, Policies>& y); template <class T, class Policies> interval<T, Policies> operator*(const interval<T, Policies>& x, const interval<T, Policies>& y); template <class T, class Policies> interval<T, Policies> operator*(const interval<T, Policies>& x, const T& y); template <class T, class Policies> interval<T, Policies> operator*(const T& x, const interval<T, Policies>& y); template <class T, class Policies> interval<T, Policies> operator/(const interval<T, Policies>& x, const interval<T, Policies>& y); template <class T, class Policies> interval<T, Policies> operator/(const interval<T, Policies>& x, const T& y); template <class T, class Policies> interval<T, Policies> operator/(const T& r, const interval<T, Policies>& x); /* algebraic functions: sqrt, abs, square, pow */ template <class T, class Policies> interval<T, Policies> abs(const interval<T, Policies>& x); template <class T, class Policies> interval<T, Policies> sqrt(const interval<T, Policies>& x); template <class T, class Policies> interval<T, Policies> square(const interval<T, Policies>& x); template <class T, class Policies> interval<T, Policies> pow(const interval<T, Policies>& x, int y); /* transcendental functions: exp, log */ template <class T, class Policies> interval<T, Policies> exp(const interval<T, Policies>& x); template <class T, class Policies> interval<T, Policies> log(const interval<T, Policies>& x); /* fmod, for trigonometric function argument reduction (see below) */ template <class T, class Policies> interval<T, Policies> fmod(const interval<T, Policies>& x, const interval<T, Policies>& y); template <class T, class Policies> interval<T, Policies> fmod(const interval<T, Policies>& x, const T& y); template <class T, class Policies> interval<T, Policies> fmod(const T& x, const interval<T, Policies>& y); /* trigonometric functions */ template <class T, class Policies> interval<T, Policies> sin(const interval<T, Policies>& x); template <class T, class Policies> interval<T, Policies> cos(const interval<T, Policies>& x); template <class T, class Policies> interval<T, Policies> tan(const interval<T, Policies>& x); template <class T, class Policies> interval<T, Policies> asin(const interval<T, Policies>& x); template <class T, class Policies> interval<T, Policies> acos(const interval<T, Policies>& x); template <class T, class Policies> interval<T, Policies> atan(const interval<T, Policies>& x); /* hyperbolic trigonometric functions */ template <class T, class Policies> interval<T, Policies> sinh(const interval<T, Policies>& x); template <class T, class Policies> interval<T, Policies> cosh(const interval<T, Policies>& x); template <class T, class Policies> interval<T, Policies> tanh(const interval<T, Policies>& x); template <class T, class Policies> interval<T, Policies> asinh(const interval<T, Policies>& x); template <class T, class Policies> interval<T, Policies> acosh(const interval<T, Policies>& x); template <class T, class Policies> interval<T, Policies> atanh(const interval<T, Policies>& x); /* min, max external functions (NOT std::min/max, see below) */ template <class T, class Policies> interval<T, Policies> max(const interval<T, Policies>& x, const interval<T, Policies>& y); template <class T, class Policies> interval<T, Policies> max(const interval<T, Policies>& x, const T& y); template <class T, class Policies> interval<T, Policies> max(const T& x, const interval<T, Policies>& y); template <class T, class Policies> interval<T, Policies> min(const interval<T, Policies>& x, const interval<T, Policies>& y); template <class T, class Policies> interval<T, Policies> min(const interval<T, Policies>& x, const T& y); template <class T, class Policies> interval<T, Policies> min(const T& x, const interval<T, Policies>& y); /* bounds-related interval functions */ template <class T, class Policies> T lower(const interval<T, Policies>& x); template <class T, class Policies> T upper(const interval<T, Policies>& x); template <class T, class Policies> T width(const interval<T, Policies>& x); template <class T, class Policies> T median(const interval<T, Policies>& x); template <class T, class Policies> T norm(const interval<T, Policies>& x); /* bounds-related interval functions */ template <class T, class Policies> bool empty(const interval<T, Policies>& b); template <class T, class Policies> bool singleton(const interval<T, Policies>& x); template <class T, class Policies> bool equal(const interval<T, Policies>& x, const interval<T, Policies>& y); template <class T, class Policies> bool in(const T& r, const interval<T, Policies>& b); template <class T, class Policies> bool in_zero(const interval<T, Policies>& b); template <class T, class Policies> bool subset(const interval<T, Policies>& a, const interval<T, Policies>& b); template <class T, class Policies> bool proper_subset(const interval<T, Policies>& a, const interval<T, Policies>& b); template <class T, class Policies> bool overlap(const interval<T, Policies>& x, const interval<T, Policies>& y); /* set manipulation interval functions */ template <class T, class Policies> interval<T, Policies> intersection(const interval<T, Policies>& x, const interval<T, Policies>& y); template <class T, class Policies> interval<T, Policies> hull(const interval<T, Policies>& x, const interval<T, Policies>& y); template <class T, class Policies> interval<T, Policies> hull(const interval<T, Policies>& x, const T& y); template <class T, class Policies> interval<T, Policies> hull(const T& x, const interval<T, Policies>& y); template <class T, class Policies> interval<T, Policies> hull(const T& x, const T& y); template <class T, class Policies> std::pair<interval<T, Policies>, interval<T, Policies> > bisect(const interval<T, Policies>& x); /* interval comparison operators */ template<class T, class Policies> bool operator<(const interval<T, Policies>& x, const interval<T, Policies>& y); template<class T, class Policies> bool operator<(const interval<T, Policies>& x, const T& y); template<class T, class Policies> bool operator<(const T& x, const interval<T, Policies>& y); template<class T, class Policies> bool operator<=(const interval<T, Policies>& x, const interval<T, Policies>& y); template<class T, class Policies> bool operator<=(const interval<T, Policies>& x, const T& y); template<class T, class Policies> bool operator<=(const T& x, const interval<T, Policies>& y); template<class T, class Policies> bool operator>(const interval<T, Policies>& x, const interval<T, Policies>& y); template<class T, class Policies> bool operator>(const interval<T, Policies>& x, const T& y); template<class T, class Policies> bool operator>(const T& x, const interval<T, Policies>& y); template<class T, class Policies> bool operator>=(const interval<T, Policies>& x, const interval<T, Policies>& y); template<class T, class Policies> bool operator>=(const interval<T, Policies>& x, const T& y); template<class T, class Policies> bool operator>=(const T& x, const interval<T, Policies>& y);

template<class T, class Policies> bool operator==(const interval<T, Policies>& x, const interval<T, Policies>& y); template<class T, class Policies> bool operator==(const interval<T, Policies>& x, const T& y); template<class T, class Policies> bool operator==(const T& x, const interval<T, Policies>& y); template<class T, class Policies> bool operator!=(const interval<T, Policies>& x, const interval<T, Policies>& y); template<class T, class Policies> bool operator!=(const interval<T, Policies>& x, const T& y); template<class T, class Policies> bool operator!=(const T& x, const interval<T, Policies>& y); namespace interval_lib { template<class T, class Policies> interval<T, Policies> division_part1(const interval<T, Policies>& x, const interval<T, Policies& y, bool& b); template<class T, class Policies> interval<T, Policies> division_part2(const interval<T, Policies>& x, const interval<T, Policies& y, bool b = true); template<class T, class Policies> interval<T, Policies> multiplicative_inverse(const interval<T, Policies>& x); template<class I> I add(const typename I::base_type& x, const typename I::base_type& y); template<class I> I sub(const typename I::base_type& x, const typename I::base_type& y); template<class I> I mul(const typename I::base_type& x, const typename I::base_type& y); template<class I> I div(const typename I::base_type& x, const typename I::base_type& y); } // namespace interval_lib } // namespace numeric } // namespace boost

`interval`

template <class T, class Policies = typename interval_lib::default_policies<T>::type> class interval { public: typedef T base_type; typedef Policies traits_type; interval(const T& v = T()); interval(const T& l, const T& u); template<class Policies2> interval(const interval<T,Policies2>& r); // compiler-generated copy constructor and assignment operator are fine interval& operator=(const T& x); void assign(const T& l, const T& u); const T& lower() const; const T& upper() const; static interval empty(); static interval whole(); static interval hull(const T& x, const T& y); interval& operator+= (const T& r); interval& operator+= (const interval& r); interval& operator-= (const T& r); interval& operator-= (const interval& r); interval& operator*= (const T& r); interval& operator*= (const interval& r); interval& operator/= (const T& r); interval& operator/= (const interval& r); };

The two main constructors accept one and two arguments of type T. The
first one produces a singleton interval [v,v]; and the second one an interval
[l,u]. The second one will use the checking policy if u<l. If you are not
sure whether the bounds are inverted or not, it is better to use the function
call `hull(l,u)`

. This will not create a problem if u<l.

There is a template constructor in order to change the traits parameter of an interval. But there is no constructor that changes the base type of an interval (it could lead to violating the inclusion property if the old base type is not contained in the new one).

There is an assign function in order to directly change the bounds of an interval. It behaves like the second constructor if the bounds are inverted. There is no assign function that directly takes an interval or only one number as a parameter; just use the assignment operator in that case.

Assignment operator, copy constructor and destructor are the ones the compiler automatically generates. There is also an assignment operator for the type T.

The static functions `empty`

and `whole`

produces
the corresponding intervals. They are static member functions rather than
global functions because they cannot guess their return types. Likewise for
`hull`

.

Some of the following functions expect `min`

and
`max`

to be defined for the base type. Those are the only
requirements for the `interval`

class (but the policies can have
other requirements).

`+`

`-`

`*`

`/`

`+=`

`-=`

`*=`

`/=`

The basic operations are the unary minus and the binary `+`

`-`

`*`

`/`

. The unary minus takes an
interval and returns an interval. The binary operations take two intervals,
or one interval and a number, and return an interval. If an argument is a
number instead of an interval, you can expect the result to be the same as if
the number was first converted to an interval. This property will be true for
all the following functions and operators.

There are also some assignment operators `+=`

`-=`

`*=`

`/=`

. There is not much to say: ```
x op=
y
```

is equivalent to `x = x op y`

. If an exception is thrown
during the computations, the l-value is not modified (but it may be corrupt
if an exception is thrown by the base type during an assignment).

The operators `/`

and `/=`

will try to produce an
empty interval if the denominator is exactly zero. If the denominator
contains zero (but not only zero), the result will be the smallest interval
containing the set of division results; so one of its bound will be infinite,
but it may not be the whole interval.

`lower`

`upper`

`median`

`width`

`norm`

`lower`

, `upper`

, `median`

respectively
compute the lower bound, the upper bound, and the median number of an
interval (`(lower+upper)/2`

rounded to nearest).
`width`

computes the width of an interval
(`upper-lower`

rounded toward plus infinity). `norm`

computes an upper bound of the interval in absolute value; it is a
mathematical norm (hence the name) similar to the absolute value for real
numbers.

`min`

`max`

`abs`

`square`

`pow`

`division_part?`

`multiplicative_inverse`

The functions `min`

, `max`

and `abs`

are
also defined. Please do not mistake them for the functions defined in the
standard library (aka `a<b?a:b`

, `a>b?a:b`

,
`a<0?-a:a`

). These functions are compatible with the elementary
property of interval arithmetic. For example, max([*a*,*b*],
[*c*,*d*]) = {max(*x*,*y*) such that *x* in
[*a*,*b*] and *y* in [*c*,*d*]}. They are not
defined in the `std`

namespace but in the boost namespace in order
to avoid conflict with the other definitions.

The `square`

function is quite particular. As you can expect
from its name, it computes the square of its argument. The reason this
function is provided is: `square(x)`

is not `x*x`

but
only a subset when `x`

contains zero. For example, [-2,2]*[-2,2] =
[-4,4] but [-2,2]˛ = [0,4]; the result is a smaller interval. Consequently,
`square(x)`

should be used instead of `x*x`

because of
its better accuracy and a small performance improvement.

As for `square`

, `pow`

provides an efficient and
more accurate way to compute the integer power of an interval. Please note:
when the power is 0 and the interval is not empty, the result is 1, even if
the input interval contains 0. `multiplicative_inverse`

computes
1/x.

The functions `division_part1`

and `division_part2`

are useful when the user expects the division to return disjoint intervals if
necessary. For example, the narrowest closed set containg [2,3] / [-2,1] is
not ]-∞,∞[ but the union of ]-∞,-1] and [2,∞[.
When the result of the division is representable by only one interval,
`division_part1`

returns this interval and sets the boolean
reference to `false`

. However, if the result needs two intervals,
`division_part1`

returns the negative part and sets the boolean
reference to `true`

; a call to `division_part2`

is now
needed to get the positive part. This second function can take the boolean
returned by the first function as last argument. If this bool is not given,
its value is assumed to be true and the behavior of the function is then
undetermined if the division does not produce a second interval.

`intersect`

`hull`

`overlap`

`in`

`in_zero`

`subset`

`proper_subset`

`empty`

`singleton`

`equal`

`intersect`

computes the set intersection of two closed sets,
`hull`

computes the smallest interval which contains the two
parameters; those parameters can be numbers or intervals. If one of the
arguments is an invalid number or an empty interval, the function will only
use the other argument to compute the resulting interval (if allowed by the
checking policy).

There is no union function since the union of two intervals is not an
interval if they do not overlap. If they overlap, the `hull`

function computes the union.

The function `overlap`

tests if two intervals have some common
subset. `in`

tests if a number is in an interval;
`in_zero`

is a variant which tests if zero is in the interval.
`subset`

tests if the first interval is a subset of the second;
and `proper_subset`

tests if it is a proper subset.
`empty`

and `singleton`

test if an interval is empty or
is a singleton. Finally, `equal`

tests if two intervals are
equal.

`sqrt`

`log`

`exp`

`sin`

`cos`

`tan`

`asin`

`acos`

`atan`

`sinh`

`cosh`

`tanh`

`asinh`

`acosh`

`atanh`

`fmod`

The functions `sqrt`

, `log`

, `exp`

,
`sin`

, `cos`

, `tan`

, `asin`

,
`acos`

, `atan`

, `sinh`

, `cosh`

,
`tanh`

, `asinh`

, `acosh`

, `atanh`

are also defined. There is not much to say; these functions extend the
traditional functions to the intervals and respect the basic property of
interval arithmetic. They use the checking policy to
produce empty intervals when the input interval is strictly outside of the
domain of the function.

The function `fmod(interval x, interval y)`

expects the lower
bound of `y`

to be strictly positive and returns an interval
`z`

such as `0 <= z.lower() < y.upper()`

and such
as `z`

is a superset of `x-n*y`

(with `n`

being an integer). So, if the two arguments are positive singletons, this
function `fmod(interval, interval)`

will behave like the
traditional function `fmod(double, double)`

.

Please note that `fmod`

does not respect the inclusion property
of arithmetic interval. For example, the result of
`fmod`

([13,17],[7,8]) should be [0,8] (since it must contain [0,3]
and [5,8]). But this answer is not really useful when the purpose is to
restrict an interval in order to compute a periodic function. It is the
reason why `fmod`

will answer [5,10].

`add`

`sub`

`mul`

`div`

These four functions take two numbers and return the enclosing interval for the operations. It avoids converting a number to an interval before an operation, it can result in a better code with poor optimizers.

Some constants are hidden in the `boost::interval_lib`

namespace. They need to be explicitely templated by the interval type. The
functions are `pi<I>()`

, `pi_half<I>()`

and
`pi_twice<I>()`

, and they return an object of interval type
`I`

. Their respective values are π, π/2 and
2π.

The interval class and all the functions defined around this class never throw any exceptions by themselves. However, it does not mean that an operation will never throw an exception. For example, let's consider the copy constructor. As explained before, it is the default copy constructor generated by the compiler. So it will not throw an exception if the copy constructor of the base type does not throw an exception.

The same situation applies to all the functions: exceptions will only be thrown if the base type or one of the two policies throws an exception.

The interval support library consists of a collection of classes that can
be used and combined to fabricate almost various commonly-needed interval
policies. In contrast to the basic classes and functions which are used in
conjunction with `interval<T>`

(and the default policies as
the implicit second template parameter in this type), which belong simply to
the namespace `boost`

, these components belong to the namespace
`boost::numeric::interval_lib`

.

We merely give the synopsis here and defer each section to a separate web page since it is only intended for the advanced user. This allows to expand on each topic with examples, without unduly stretching the limits of this document.

namespace boost { namespace numeric { namespace interval_lib { /* built-in rounding policy and its specializations */ template <class T> struct rounded_math; template <> struct rounded_math<float>; template <> struct rounded_math<double>; template <> struct rounded_math<long double>; /* built-in rounding construction blocks */ template <class T> struct rounding_control; template <class T, class Rounding = rounding_control<T> > struct rounded_arith_exact; template <class T, class Rounding = rounding_control<T> > struct rounded_arith_std; template <class T, class Rounding = rounding_control<T> > struct rounded_arith_opp; template <class T, class Rounding> struct rounded_transc_dummy; template <class T, class Rounding = rounded_arith_exact<T> > struct rounded_transc_exact; template <class T, class Rounding = rounded_arith_std <T> > struct rounded_transc_std; template <class T, class Rounding = rounded_arith_opp <T> > struct rounded_transc_opp; template <class Rounding> struct save_state; template <class Rounding> struct save_state_nothing; /* built-in checking policies */ template <class T> struct checking_base; template <class T, class Checking = checking_base<T>, class Exception = exception_create_empty> struct checking_no_empty; template <class T, class Checking = checking_base<T> > struct checking_no_nan; template <class T, class Checking = checking_base<T>, class Exception = exception_invalid_number> struct checking_catch_nan; template <class T> struct checking_strict; /* some metaprogramming to manipulate interval policies */ template <class Rounding, class Checking> struct policies; template <class OldInterval, class NewRounding> struct change_rounding; template <class OldInterval, class NewChecking> struct change_checking; template <class OldInterval> struct unprotect; /* constants, need to be explicitly templated */ template<class I> I pi(); template<class I> I pi_half(); template<class I> I pi_twice(); /* interval explicit comparison functions: * the mode can be cer=certainly or pos=possibly, * the function lt=less_than, gt=greater_than, le=less_than_or_equal_to, ge=greater_than_or_equal_to * eq=equal_to, ne= not_equal_to */ template <class T, class Policies> bool cerlt(const interval<T, Policies>& x, const interval<T, Policies>& y); template <class T, class Policies> bool cerlt(const interval<T, Policies>& x, const T& y); template <class T, class Policies> bool cerlt(const T& x, const interval<T, Policies>& y); template <class T, class Policies> bool cerle(const interval<T, Policies>& x, const interval<T, Policies>& y); template <class T, class Policies> bool cerle(const interval<T, Policies>& x, const T& y); template <class T, class Policies> bool cerle(const T& x, const interval<T, Policies>& y); template <class T, class Policies> bool cergt(const interval<T, Policies>& x, const interval<T, Policies>& y); template <class T, class Policies> bool cergt(const interval<T, Policies>& x, const T& y); template <class T, class Policies> bool cergt(const T& x, const interval<T, Policies>& y); template <class T, class Policies> bool cerge(const interval<T, Policies>& x, const interval<T, Policies>& y); template <class T, class Policies> bool cerge(const interval<T, Policies>& x, const T& y); template <class T, class Policies> bool cerge(const T& x, const interval<T, Policies>& y); template <class T, class Policies> bool cereq(const interval<T, Policies>& x, const interval<T, Policies>& y); template <class T, class Policies> bool cereq(const interval<T, Policies>& x, const T& y); template <class T, class Policies> bool cereq(const T& x, const interval<T, Policies>& y); template <class T, class Policies> bool cerne(const interval<T, Policies>& x, const interval<T, Policies>& y); template <class T, class Policies> bool cerne(const interval<T, Policies>& x, const T& y); template <class T, class Policies> bool cerne(const T& x, const interval<T, Policies>& y); template <class T, class Policies> bool poslt(const interval<T, Policies>& x, const interval<T, Policies>& y); template <class T, class Policies> bool poslt(const interval<T, Policies>& x, const T& y); template <class T, class Policies> bool poslt(const T& x, const interval<T, Policies>& y); template <class T, class Policies> bool posle(const interval<T, Policies>& x, const interval<T, Policies>& y); template <class T, class Policies> bool posle(const interval<T, Policies>& x, const T& y); template <class T, class Policies> bool posle(const T& x, const interval<T, Policies>& y); template <class T, class Policies> bool posgt(const interval<T, Policies>& x, const interval<T, Policies>& y); template <class T, class Policies> bool posgt(const interval<T, Policies>& x, const T& y); template <class T, class Policies> bool posgt(const T& x, const interval<T, Policies> & y); template <class T, class Policies> bool posge(const interval<T, Policies>& x, const interval<T, Policies>& y); template <class T, class Policies> bool posge(const interval<T, Policies>& x, const T& y); template <class T, class Policies> bool posge(const T& x, const interval<T, Policies>& y); template <class T, class Policies> bool poseq(const interval<T, Policies>& x, const interval<T, Policies>& y); template <class T, class Policies> bool poseq(const interval<T, Policies>& x, const T& y); template <class T, class Policies> bool poseq(const T& x, const interval<T, Policies>& y); template <class T, class Policies> bool posne(const interval<T, Policies>& x, const interval<T, Policies>& y); template <class T, class Policies> bool posne(const interval<T, Policies>& x, const T& y); template <class T, class Policies> bool posne(const T& x, const interval<T, Policies>& y); /* comparison namespaces */ namespace compare { namespace certain; namespace possible; namespace lexicographic; namespace set; namespace tribool; } // namespace compare } // namespace interval_lib } // namespace numeric } // namespace boost

Each component of the interval support library is detailed in its own page.

One of the biggest problems is problably the correct use of the comparison
functions and operators. First, functions and operators do not try to know if
two intervals are the same mathematical object. So, if the comparison used is
"certain", then `x != x`

is always true unless `x`

is a
singleton interval; and the same problem arises with `cereq`

and
`cerne`

.

Another misleading interpretation of the comparison is: you cannot always expect [a,b] < [c,d] to be !([a,b] >= [c,d]) since the comparison is not necessarily total. Equality and less comparison should be seen as two distincts relational operators.

This problem is a corollary of the previous problem with ```
x !=
x
```

. All the functions of the library only consider the value of an
interval and not the reference of an interval. In particular, you should not
expect (unless `x`

is a singleton) the following values to be
equal: `x/x`

and 1, `x*x`

and `square(x)`

,
`x-x`

and 0, etc. So the main cause of wide intervals is that
interval arithmetic does not identify different occurences of the same
variable. So, whenever possible, the user has to rewrite the formulas to
eliminate multiple occurences of the same variable. For example,
`square(x)-2*x`

is far less precise than
`square(x-1)-1`

.

As explained in this section, a good way to speed up computations when the base type is a basic floating-point type is to unprotect the intervals at the hot spots of the algorithm. This method is safe and really an improvement for interval computations. But please remember that any basic floating-point operation executed inside the unprotection blocks will probably have an undefined behavior (but only for the current thread). And do not forget to create a rounding object as explained in the example.

The purpose of this library is to provide an efficient and generalized way
to deal with interval arithmetic through the use of a templatized class
`boost::interval`

. The big contention for which we provide a
rationale is the format of this class template.

It would have been easier to provide a class interval whose base type is
double. Or to follow `std::complex`

and allow only specializations
for `float`

, `double`

, and `long double`

. We
decided not to do this to allow intervals on custom types, e.g.
fixed-precision bigfloat library types (MPFR, etc.)

**Policy design.** Although it was tempting to make it a
class template with only one template argument, the diversity of uses for an
interval arithmetic practically forced us to use policies. The behavior of
this class can be fixed by two policies. These policies are packaged into a
single policy class, rather than making `interval`

with three
template parameters. This is both for ease of use (the policy class can be
picked by default) and for readability.

The first policy provides all the mathematical functions on the base type needed to define the functions on the interval type. The second one sets the way exceptional cases encountered during computations are handled.

We could foresee situations where any combination of these policies would
be appropriate. Moreover, we wanted to enable the user of the library to
reuse the `interval`

class template while at the same time
choosing his own behavior. See this page for some
examples.

**Rounding policy.** The library provides specialized
implementations of the rounding policy for the primitive types float and
double. In order for these implementations to be correct and fast, the
library needs to work a lot with rounding modes. Some processors are directly
dealt with and some mecanisms are provided in order to speed up the
computations. It seems to be heavy and hazardous optimizations for a gain of
only a few computer cycles; but in reality, the speed-up factor can easily go
past 2 or 3 depending on the computer. Moreover, these optimizations do not
impact the interface in any major way (with the design we have chosen,
everything can be added by specialization or by passing different template
parameters).

**Pred/succ.** In a previous version, two functions
`pred`

and `succ`

, with various corollaries like
`widen`

, were supplied. The intent was to enlarge the interval by
one ulp (as little as possible), e.g. to ensure the inclusion property. Since
making interval a template of T, we could not define *ulp* for a random
parameter. In turn, rounding policies let us eliminate entirely the use of
ulp while making the intervals tighter (if a result is a representable
singleton, there is no use to widen the interval). We decided to drop those
functions.

**Specialization of std::less.** Since the
operator

`<`

depends on the comparison namespace locally chosen
by the user, it is not possible to correctly specialize
`std::less`

. So you have to explicitely provide such a class to
all the algorithms and templates that could require it (for example,
`std::map`

).**Input/output.** The interval library does not include I/O
operators. Printing an interval value allows a lot of customization: some
people may want to output the bounds, others may want to display the median
and the width of intervals, and so on. The example file io.cpp```
shows some possibilities and may serve as a foundation in order for the user
to define her own operators.
```

**Conversion from interval<T1> to
interval<T>.** There is no such conversion in the
library. Specifically, there is no way to change the base number type without
explicitly casting both the lower and upper bounds from

`T1`

to
`T2`

and using a constructor. We did not want to provide this
conversion because there is no easy way to make sure the inclusion property
is guaranteed: a simple casting with loss of precision can lead to invalid or
empty intervals. Moreover it is not clear who should be responsible for the
conversion: `T2`

is not required to know about all the possible
types `T1`

. We also do not want to throw an exception is there is
a loss of precision (as is done for instance in `numeric_cast`

);
instead, the result should be correctly rounded to ensure the inclusion
property. So we decided not to provide the conversion and leave it up to the
user to do it himself or herself.**Mixed operations with integers.** When using and reusing
template codes, it is common there are operations like `2*x`

.
However, the library does not provide them by default because the conversion
from `int`

to the base number type is not always correct. So the
functions have been put in a separate header and the user need to include
them explicitely if she wants to benefit from these mixed operators Another
point, there is no mixed comparison operators due to the technical way they
are defined.

This library was mostly inspired by previous work from Jens Maurer. Some discussions about his work are reproduced here and the work itself can be found here. Jeremy Siek and Maarten Keijzer provided some rounding control for MSVC and Sparc platforms.

Guillaume Melquiond, Hervé Brönnimann and Sylvain Pion started from the library left by Jens and added the policy design. Guillaume and Sylvain worked hard on the code, especially the porting and mostly tuning of the rounding modes to the different architectures. Guillaume mostly did most of the coding, while Sylvain and Hervé have provided some useful comments in order for this library to be written. Hervé reorganized and wrote chapters of the documentation based on Guillaume's great starting point.

Revised: 2003-08-16

Copyright (c) Guillaume Melquiond, Sylvain Pion, Hervé Brönnimann, 2002.
Polytechnic University.

Copyright (c) Guillaume Melquiond, 2003.