LEDA Guide: Algebraic Real Numbers

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Algebraic Real Numbers

The data type real approximates mathematics' real numbers. It most powerful number type.

Example

The following program shows a computation where the exact real arithmetic evaluates to zero in contrast to floating point arithmetic. Notice that the explicit cast of the numbers 12 and 17 is really necessary, here. Otherwise the right hand side is evaluated with double arithmetic.

#include <LEDA/numbers/real.h>

using namespace leda;

int main()
{
  real x;
  x=(sqrt(real(17))-sqrt(real(12)));
  x*=(sqrt(real(17))+sqrt(real(12)));
  x-=5;
  if (x==0) std::cout << "x==0\n";
  else std::cout << "x!=0\n";

  double xd;
  xd=(sqrt(17)-sqrt(12));
  xd*=(sqrt(17)+sqrt(12));
  xd-=5;
  if (xd==0) std::cout << "xd==0\n";
  else std::cout << "xd!=0\n";

  return 0;
}

Strengths

computations with reals always yield mathematically correct results
can be used like double and together with built-in number types
many powerful arithmetic operations predefined, like square roots and k-th roots
efficient implementation

Disadvantages

space grows linearly with the size of a computation (all operations of a computation are recorded)
about 10-80 times slower than double
in general less efficient than Integers of Arbitrary Length and Rational Numbers. (This disadvantage is partially offset by filtering mechanisms.)

Tip

Use reals to do exact computations with square roots and k-th roots. Otherwise consider using Big Floatingpoint Numbers.

Algebraic Real Numbers

Example

Strengths

Disadvantages

Tip

See also: