Introduction

Traditional representations of real numbers reach only a restricted subset of reals (rationals, algebraic). Other disadvantages are:

1. many computations lead outside these subsets;
2. some operations are expensive in these representations (see [(Roy, Szpirglas, 90)]);
3. other operations are numerically unsafe.

Exact real computation reaches all computable numbers (see [(Boehm, Cartwright, 90),(Gianantonio, 93),(Escardo, 2000),(Menissier-Morain, 2000)]). It is also numerically safe.

Among the possible representations, we mention:

• lazy infinite sequences of rational numbers;
• procedures producing arbitrarily precise rational approximations on demand.

In traditional (floating point-based) numeric computation it is often difficult to ensure safety, i.e. to provide guaranteed error bounds that are sufficiently small (see [(Schewchuk, 97)]).

The representation used in exact real computation is mathematically consistent: the models behave exactly like the mathematical objects which they represent.

Thus, it is plausible that exact real computation:

• is sometimes much faster than traditional symbolic computation,
• is sometimes much easier to analyze/interpret than numerical computation.

Task:  Provide a package for manipulating objects that represent computable real or complex numbers, or vectors/matrices with real or complex entries.

Our belief is that exact real computation can be realized within existing computer algebra systems, because they contain most of the necessary ingredients:

1. high level algorithms using arbitrary precision arithmetic and giving guaranteed error bounds,
2. conversion between numerical and symbolic data.

For many problems, it suffices to provide small wrappers encoding the necessary information about the error propagation. often this information is already available in the literature (see e.g. [(Higham, 96)]).

This research has been supported by the Austrian Science Foundation FWF in the frame of the project SFB 013.