Here, boldface denotes intervals, lower case denotes
scalar quantities, and upper case denotes vectors and matrices.
Brackets "
" delimit intervals while parentheses
"
" delimit vectors and matrices. Underscores
denote lower bounds of intervals and overscores denote upper
bounds of intervals. Corresponding lower case letters denote
components of vectors. The set of real intervals is denoted by
. Interval vectors will also be called boxes.
If 
and 
, then the four elementary
operations for idealized interval arithmetic obey
Thus, the image of each of the four basic interval operations is the exact range of the corresponding real operation. Although Equation (3) characterizes these operations mathematically, interval arithmetic's usefulness is due to the operational definitions. For example,
The ranges of the four elementary interval arithmetic
operations are exactly the ranges of the corresponding real
operations. If such operations are composed, bounds on
the ranges of real functions can be obtained. For example,
if
then
which contains the exact range [-1/4,0]. (This is
necessarily so.)
Such bounds on ranges can be used in place of Lipschitz constants.
In fact, bounds from
interval arithmetic often are sharper, and are simpler to derive than
bounds from other techniques. For example, if f is as in
Equation (9), then the mean value theorem
gives 
for some unknown 
between
0.5 and x. If 
, then may be replaced by
the interval [0,1] to obtain 
.
This leads to a second set of bounds on the range of f:
for . (Intersecting these two ranges gives
, a sharper bound than
either of them individually.)
The power of interval arithmetic lies in its implementation on computers. In particular, outwardly rounded interval arithmetic allows rigorous enclosures for the ranges of operations and functions. This makes a qualitative difference in scientific computations, since the results are now intervals in which the exact result must lie. It also enables use of computations for automated theorem proving.
Directed rounding proceeds as follows. Much modern computing equipment
(including machines that support IEEE standard arithmetic
[16], such as most PC's and workstations)
allows the result of an arithmetic operation to be rounded down
to the nearest machine number less than the mathematically correct
result, rounded up to the nearest machine number greater than
or equal to the mathematically correct result, or rounded to the
machine number nearest to the mathematically correct result. For
example, take 
. If 
is
rounded down after computation and 
is rounded up after
computation, then the resulting interval 
that is
represented in the machine must contain the exact range of
x+y for 
and 
.
Good enclosures for the ranges of transcendental functions
such as "
" and ``
" can be computed. Thus, interval
arithmetic can be carried out for virtually any expression that
can be evaluated with floating point arithmetic. However, since
interval arithmetic is only subdistributive, expressions
that are equivalent in real arithmetic differ in interval arithmetic.
In particular, computations should be arranged so that overestimation
of ranges is minimized. The fact that naively arranged computations
do not always give adequately narrow bounds on the range has been
the source of controversy, but there have been advances in recent
years in the astute use of interval arithmetic.
Additional introductory and advanced details and explanations of interval arithmetic can be found in the books [1, 2, 3, 4, 5, 6, 8, 10, 11, 12, 13, 14]. A World-Wide-Web entry point for interval computations is http://cs.utep.edu/interval-comp/main.html
