A Fast Computation of the Best $${k}$$ k -Digit Rational Approximation to a Real Number

Abstract

Given a real number α, we aim at computing the best rational approximation with at most k digits and with exactly k digits at the numerator (denominator). Our approach exploits Farey sequences. Our method turns out to be very fast in the sense that, once the development of α in continued fractions is available, the required operations are just a few and their number remains essentially constant for any k (in double precision finite arithmetic). Estimations of error bounds are also provided.

Birkhoff Averages and Rotational Invariant Circles for Area-Preserving Maps

“…Algorithm 2 ignores issues of finite precision arithmetic, and is not efficient if the Farey path has a long string of repeated symbols. An algorithm that does not have this deficit is given in [CP16].…”