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

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Abstract

Given a real number \({\alpha}\), 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 \({\alpha}\) 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.

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  1. Department of Mathematics, Politecnico di Milano, Milan, Italy

    Maurizio Citterio & Raffaella Pavani

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  1. Maurizio Citterio
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  2. Raffaella Pavani
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Correspondence to Raffaella Pavani.

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Citterio, M., Pavani, R. A Fast Computation of the Best \({k}\)-Digit Rational Approximation to a Real Number. Mediterr. J. Math. 13, 4321–4331 (2016). https://doi.org/10.1007/s00009-016-0747-z

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  • DOI: https://doi.org/10.1007/s00009-016-0747-z

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