Introduction

All over the world numerous computers are used for real number computation. They evaluate real functions, find zeroes of functions, determine eigenvalues and integrals and solve differential equations. They perform, or at least are expected to perform, computations on sets like ℝ (the set of real numbers), ℝn, O(ℝ) (the open subsets of real numbers), K(ℝn) (the compact subsets of ℝn) or C[O; 1] (the continuous functions from the real unit interval to the real numbers). The increasing demand for reliable as well as fast software in scientific computation and engineering requires a sound and broad foundation. We agree with L. Blum et al. [BCSS96] (also in [BCSS98], however, see Sect. 9.7):