# Approximate rational dependence

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

$q_1\alpha+q_2\beta+q_3\gamma+q_4\delta\approx0$

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

What is such an algorithm called?

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I think the LLL-algorithm of Lenstra, Lenstra, Lovasz does something like this –  Cocopuffs Aug 11 '12 at 8:49
@Cocopuffs: Can this algorithm handle rational numbers or is it integer only? –  Gerenuk Aug 11 '12 at 8:55
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. –  Cocopuffs Aug 11 '12 at 9:44
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). –  Raymond Manzoni Aug 11 '12 at 10:56