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

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.