Exact real computer arithmetic with continued fractions

Author

Vuillemin, Jean E.

Author_Institution

Centre de Recherche de Paris, France

Volume

39

Issue

8

fYear

1990

fDate

8/1/1990 12:00:00 AM

Firstpage

1087

Lastpage

1105

Abstract

A representation of the computable real numbers by continued fractions is introduced. This representation deals with the subtle points of undecidable comparison and integer division, as well as representing the infinite 1/0 and undefined 0/0 numbers. Two general algorithms for performing arithmetic operations are introduced. The algebraic algorithm, which computes sums and products of continued fractions as a special case, basically operates in a positional manner, producing one term of output for each term of input. The transcendental algorithm uses a general formula of Gauss to compute the continued fractions of exponentials, logarithms, trigonometric functions, and a wide class of special functions. A prototype system has been implemented in LeLisp and the performance of these algorithms is promising

Keywords

digital arithmetic; number theory; Gauss; LeLisp; algebraic algorithm; arithmetic operations; computable real numbers; continued fractions; exact real computer arithmetic; exponentials; infinite 1/0; integer division; logarithms; positional; products; special functions; sums; transcendental algorithm; trigonometric functions; undecidable comparison; undefined 0/0 numbers; Buildings; Digital arithmetic; Gaussian processes; Geometry; H infinity control; Heart; Prototypes; Trademarks;

fLanguage

English

Journal_Title

Computers, IEEE Transactions on

Publisher

ieee

ISSN

0018-9340

Type

jour

DOI

10.1109/12.57047

Filename

57047