Version 5.0 The LEDA User Manual

5.5 Interval Arithmetic in LEDA ( interval )
1. Definition
An instance of the data type interval represents a real interval I = [a, b]. The basic
interval operations +, −, ∗, /, √ are available. Type interval can be used to approximate
exact real arithmetic operations by inexact interval operations, as follows. Each input
number x i is converted into the interval {x i } and all real operations are replaced by
interval operations. If x is the result of the exact real calculation and I the interval
computed by type interval, it is guaranteed that I contains x. I can be seen as a more or
less accurate approximation of x. In many cases the computed interval I is small enough
to provide a useful approximation of x and the exact sign of x. There are four different
implementations of intervals (consult the implementation section below for details):
• Class interval bound absolute
• Class interval bound relative
• Class interval round inside
• Class interval round outside, which is usually the fastest but requires that the
IEEE754 rounding mode ieee positive is activated, e.g. by using the LEDA class
fpu.
The interface of all interval variants are identical. However, note that the types
interval round inside and interval round outside are only available on some explicitly supported
UNIX platforms, currently including SPARC, MIPS, i386 (PC’s compatible to
80386 or higher), and ALPHA. For all platforms, the name interval stands for the default
implementation interval bound absolute.
#include < LEDA/numbers/interval.h >
interval x; creates an instance x of type interval and initializes it with the
interval {0}
interval
x(VOLATILE I double a);
creates an instance x of type interval and initializes it with {a}
interval x(int a); creates an instance x of type interval and initializes it with {a}
interval x(long a); creates an instance x of type interval and initializes it with {a}
interval
x(const integer& a);
creates an instance x of type interval and initializes it with the
smallest possible interval containing a