After seeing question Why is $10\frac{\exp(\pi)-\log 3}{\log 2}$ almost an integer? I wonder if there is an algorithm that can find approximate rational dependence?! I pick any irrational numbers $\alpha$, $\beta$, $\gamma$, $ \delta$ and the task is to find successively good approximations for rational numbers $q_i$ in


Basically it's like continued fraction approximations, but for more irrational numbers.

What is such an algorithm called?

  • 1
    $\begingroup$ I think the LLL-algorithm of Lenstra, Lenstra, Lovasz does something like this $\endgroup$ – Cocopuffs Aug 11 '12 at 8:49
  • $\begingroup$ @Cocopuffs: Can this algorithm handle rational numbers or is it integer only? $\endgroup$ – Gerenuk Aug 11 '12 at 8:55
  • $\begingroup$ I doubt there's any optimality for rational numbers, although you can certainly divide by a common denominator. I'm not sure exactly how this would work. The problem as given I don't fully understand either, as you can obviously take $q_1 = ... = q_4 = 0$, but I'm sure you want to exclude that. $\endgroup$ – Cocopuffs Aug 11 '12 at 9:44
  • $\begingroup$ like Cocopuffs and Gerry(+1) explained you may use LLL and PSLQ. This was tried here (to get better approximations you need only to increase the precision). $\endgroup$ – Raymond Manzoni Aug 11 '12 at 10:56

It's called an integer relation algorithm. In addition to LLL mentioned in the comments, there's also PSLQ and others. The Wikipedia article on integer relation algorithms will get you started on understanding these algorithms.


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