Version 5.0 The LEDA User Manual

5.4 The data type real ( real )
1. Definition
An instance x of the data type real is a real algebraic number. There are many ways
to construct a real: either by conversion from double, bigfloat, integer or rational, by
applying one of the arithmetic operators +, −, ∗, / or d √ to real numbers or by using the
⋄-operator to define a real root of a polynomial over real numbers. One may test the
sign of a real number or compare two real numbers by any of the comparison relations
=, ≠, and ≥. The outcome of such a test is mathematically exact. We give
consider an example expression to clarify this:
x := ( √ 17 − √ 12) ∗ ( √ 17 + √ 12) − 5
Clearly, the value of x is zero. But if you evaluate x using double arithmetic you obtain
a tiny non-zero value due to rounding errors. If the data type real is used to compute
x then sign(x) yields zero. 1 There is also a non–standard version of the sign function:
the call x.sign(integer q) computes the sign of x under the precondition that |x| ≤ 2 −q
implies x = 0. This version of the sign function allows the user to assist the data type in
the computation of the sign of x, see the example below.
There are several functions to compute approximations of reals. The calls x.to bigfloat( )
and x.get bigfloat error( ) return bigfloats xnum and xerr such that |xnum − x| ≤
xerr. The user may set a bound on xerr. More precisely, after the call
x.improve approximation to(integer q) the data type guarantees xerr ≤ 2 −q . One can
also ask for double approximations of a real number x. The calls x.to double( ) and
x.get double error( ) return doubles xnum and xerr such that |xnum − x| ≤ xerr. Note
that xerr = ∞ is possible.
#include < LEDA/numbers/real.h >
2. Types
typedef polynomial Polynomial
the polynomial type.
3. Creation
reals may be constructed from data types double, bigfloat, long, int and integer. The
default constructor real( ) initializes the real to zero.
4. Operations
double x.to double( ) returns the current double approximation of x.
bigfloat x.to bigfloat( ) returns the current bigfloat approximation of x.