Calculation of Rational Numbers in an Interval Whose Denominator is the Smallest by using FP Interval Arithmetic

Abstract

The continued fraction expansion method is a fast solver to find a rational number in a given real interval whose denominator is the smallest. A simple implementation of the CF expansion method which uses floating point numbers as real numbers has a possibility to give a wrong answer by the effect of numerical round-off errors. In this paper, we show a modification of the algorithm of the CF expansion method so that it uses floating point (FP) intervals as replacements of real numbers. By this modified algorithm, the answer is obtained only when its correctness is guaranteed and the possibility to give a wrong answer is eliminated.