# Roundoff-Free Number Fields for Interval Computations

Roundoff-Free Number Fields for Interval Computations Roundoff errors are inevitable if the exact result of some operation is not representable in a computer, and has, therefore, to be approximated. To avoid roundoff errors, it is hence necessary to choose a set of computer-representable numbers in such a way that the results of all basic operations will be still in this set. In this paper, we prove that if we include arithmetic operations and computing the interval range into this operations list, then the set F of numbers will be roundoff-free iff F is a real closed field; therefore, the smallest such set is the set of all real algebraic numbers (i.e., solutions of polynomial equations with rational coefficients). http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Reliable Computing Springer Journals

# Roundoff-Free Number Fields for Interval Computations

, Volume 7 (1) – Oct 3, 2004
8 pages

/lp/springer_journal/roundoff-free-number-fields-for-interval-computations-QbKaHmAraB
Publisher
Subject
Mathematics; Numeric Computing; Approximations and Expansions; Computational Mathematics and Numerical Analysis; Mathematical Modeling and Industrial Mathematics
ISSN
1385-3139
eISSN
1573-1340
D.O.I.
10.1023/A:1011483201932
Publisher site
See Article on Publisher Site

### Abstract

Roundoff errors are inevitable if the exact result of some operation is not representable in a computer, and has, therefore, to be approximated. To avoid roundoff errors, it is hence necessary to choose a set of computer-representable numbers in such a way that the results of all basic operations will be still in this set. In this paper, we prove that if we include arithmetic operations and computing the interval range into this operations list, then the set F of numbers will be roundoff-free iff F is a real closed field; therefore, the smallest such set is the set of all real algebraic numbers (i.e., solutions of polynomial equations with rational coefficients).

### Journal

Reliable ComputingSpringer Journals

Published: Oct 3, 2004

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