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Given a real number $x$ and a limit $n$ , we can efficiently find integers $a,b$, such that $|ax+b|$ is as small as possible , if we know $|a|,|b|\le n$.

Those numbers can be determined with the convergents or semiconvergents of continued fractions.

Is this also possible for $|ax^2+bx+c|$ ? Concretely, given a real number $x$ and $|a|,|b|,|c|\le n$, can we efficiently find $a,b,c$, such that $|ax^2+bx+c|$ is as small as possible ? If not, can we at least determine easily whether a given triple $a,b,c$ is optimal ?

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    What do you mean by efficient? If we are looking for three integers in ${-n,\dots,n}$, there are only $(2n)^3$ combinations, which is polynomial time and typically considered efficient. – 2017-01-27
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    I am looking for a method similar to the continued-fraction-method, which is much faster than $O(n^3)$ – 2017-01-27

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