Issue No. 08 - August (1990 vol. 39)

ISSN: 0018-9340

pp: 1087-1105

DOI Bookmark: http://doi.ieeecomputersociety.org/10.1109/12.57047

ABSTRACT

<p>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.</p>

INDEX TERMS

exact real computer arithmetic; continued fractions; computable real numbers; undecidable comparison; integer division; infinite 1/0; undefined 0/0 numbers; arithmetic operations; algebraic algorithm; sums; products; positional; transcendental algorithm; Gauss; exponentials; logarithms; trigonometric functions; special functions; LeLisp; digital arithmetic; number theory.

CITATION

J.E. Vuillemin, "Exact Real Computer Arithmetic with Continued Fractions",

*IEEE Transactions on Computers*, vol. 39, no. , pp. 1087-1105, August 1990, doi:10.1109/12.57047