Guest editors’ introduction: Special issue on practical development of exact real number computation

Abstract

In contrast to programming with integers, where each number can be represented by a finite amount of digits, programs on real numbers have to deal with the infinite number of bits that are necessary to depict a real number in conventional representations. So the notion of ‘exactness’, which is almost trivial in the integer case, is essential for real number computations. The topic of this special issue is to present different approaches to the exactness problem. Four papers deal with arithmetic on the full set of (computable) real numbers, the other three cover important subsets: Geometric computations on algebraic numbers, floating point arithmetic with variable precision, and fixed precision floating point arithmetic enhanced with computer algebra. The first paper, by Hans-J. Boehm, gives implementational details from his Java package for constructive recursive real numbers. It shows how a data structure can be build in a ‘conventional’ programming language that preserves all of the information on numbers through their construction. Additionally, the process of the evaluation of these structures is addressed. The methods for the evaluation are the main topic of the second paper: The design of the underlying algorithms (together with proofs of correctness) is addressed by Valerie Menissier-Morain. She gives an insight on how the precision of approximations can be traced during the computation of many basic real functions. As a special example of deeply nested iterative computations, Jens Blanck considers methods for computation of iterated maps, especially the logistic map. Again, the evolution of the error terms is the main concern in this third paper.