Computer arithmetic and ill-conditioned algebraic problems

Interval arithmetic, i.e. the computation with numbers which are afflicted with tolerances, always provides reliable statements when applied in numerical algorithms on computers. It guarantees that the exact result of an algorithm lies within the computed tolerance bounds. In ill-conditioned cases these bounds may be extremly wide and although the statement still remains valid, it is in practice worthless. Methods which have been recently introduced as E-methods provide the possibility of successively diminishing the tolerance. Furthermore, the existence and uniqueness of the solution is proved (in a mathematical sense) by the computer. These methods combine the concepts of interval analysis with the computer arithmetic defined by Kulisch and Miranker. They are based on fixed point theorems and an exact scalar product is essential for their implementation.