An Improved Algorithm for Recovering Exact Value from its Approximation

Abstract

Numerical approximate computation can solve large and complex problems fast. It has the advantage of high efficiency. However it only gives approximate results, whereas we need exact results in many fields. There is a gap between approximate computation and exact results. A bridge overriding the gap was built by Zhang, in which an exact rational number is recovered from its approximation by continued fraction method when the error is less than $1/((2N+2)(N-1)N)$, where $N$ is a bound on absolute value of denominator of the rational number. In this paper, an improved algorithm is presented by which a exact rational number is recovered when the error is less than $1/(4(N-1)N)$.