Real Computations with Fake Numbers

Computations with Eeective Real Numbers

A real number x is said to be eeective if there exists an algorithm which, given a required tolerance " 2 Z 2 Z , returns a binary approximation x ~ 2 Z 2 Z for x with jx ~ ? xj < ". In this paper, we review several techniques for computations with such numbers, and we will present some new ones.

Real Fake Real Fake

Computations with real algebraic numbers of degree up to 4

Our motivation comes from computer-aided design and geometric modeling, where algorithms and predicates rely on computations of real algebraic numbers (usually of small degree). Predicates must be decided exactly in all cases, including degeneracies. In computational geometry there is a growing need for robust manipulation of curved objects (see [2] for a modern approach), since the critical po...

Computations with one and two real algebraic numbers

We present algorithmic and complexity results concerning computations with one and two real algebraic numbers, as well as real solving of univariate polynomials and bivariate polynomial systems with integer coefficients using Sturm-Habicht sequences. Our main results, in the univariate case, concern the problems of real root isolation (Th. 19) and simultaneous inequalities (Cor. 26) and in the ...

Computations with Large Numbers

Since any personal computer has a limited range of integer values, therefore, it will result in an integer overflow when a program tries to compute a value larger than machine’s maximum value. We will discuss a workable algorithm that will be able to deal with any large numbers without getting an integer overflow.