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I try, without success, to create an algorithm that can compute the average greatest common divisor of a series of integers.
For example, I have the following numbers:

  399, 710, 105, 891, 402, 102, 397, ...  

As you can see, the average gcd is approximately 100, but how to compute it ?

more details:

I try to find the carrier signal length of a HF signal. This signal is an alternation of high levels and low levels.
eg. ----__------____------__-- ...
I have the duration of each level, but this time is not accurate.
My aim is to find as quick as possible the base time of the signal.
My first idea was to compute the gcd of the first times I get, to find the carrier of the signal. But I cannot use the classical gcd because the values are not very accurate.
With a perfect signal I would have gcd(400, 700, 100, 900, 400, 100, 400) = 100

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  • $\begingroup$ What is the average gcd? 397 is a prime, so its gcd with any of the other numbers in the series is 1. $\endgroup$
    – almagest
    Jun 21, 2016 at 13:24
  • $\begingroup$ Does computing all gcds then taking the average not work..? $\endgroup$ Jun 21, 2016 at 13:25
  • $\begingroup$ @MattSamuel, all gcds give very low values, eg. (399, 710) => 1, etc..., then the average will be far of the expected value $\endgroup$ Jun 21, 2016 at 13:28
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    $\begingroup$ @Soubok: No matter which word you use, it looks like you will have to describe what it is you want, with greater detail and specificity than just choosing a word to use for it. $\endgroup$ Jun 21, 2016 at 13:46
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    $\begingroup$ Try to reformulate then, it might be an interesting problem. You'll need some kind of norm or measure, like the maximum of $\gcd(a_1+e_1,\dots,a_n+e_n)$ where $e_1+\cdots e_n<N$... $\endgroup$
    – Lehs
    Jun 21, 2016 at 14:25

3 Answers 3

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I made a similar question here, where I propose a partial solution.

How to find the approximate basic frequency or GCD of a list of numbers?

In summary, I came with this

  • being $v$ the list $\{v_1, v_2, \ldots, v_n\}$,
  • $\operatorname{mean}_{\sin}(v, x)$ $= \frac{1}{n}\sum_{i=1}^n\sin(2\pi v_i/x)$
  • $\operatorname{mean}_{\cos}(v, x)$ $= \frac{1}{n}\sum_{i=1}^n\cos(2\pi v_i/x)$
  • $\operatorname{gcd}_{appeal}(v, x)$ = $1 - \frac{1}{2}\sqrt{\operatorname{mean}_{\sin}(v, x)^2 + (\operatorname{mean}_{\cos}(v, x) - 1)^2}$

And the goal is to find the $x$ which maximizes the $\operatorname{gcd}_{appeal}$. Using the formulas and code described there, using CoCalc/Sage you can experiment with them and, in the case of your example, find that the optimum GCD is ~100.18867794375123:

testSeq = [399, 710, 105, 891, 402, 102, 397]
gcd = calculateGCDAppeal(x, testSeq)
find_local_maximum(gcd,90,110)
plot(gcd,(x, 10, 200), scale = "semilogx")

plot of GCD appeal

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One approach: take the minimum of all your numbers, here $102$ as the first trial. Divide it into all your other numbers, choosing the quotient that gives the remainder of smallest absolute value. For your example, this would give $-9,2,3,-27,-6,0,-11$ The fact that your remainders are generally negative says your divisor is too large, so try a number a little smaller. Keep going until the remainders get positive and larger. Depending on how the errors accumulate, you might also add up all the numbers and assume the sum is a multiple of the minimum interval. Here your numbers add to $3006$, so you might think this is $30$ periods of $100.2$ Are your periods constrained to integers?

If you have a stubbornly large remainder, you can think that the smallest number is not one interval but two. You might have a number around $150$, so the fundamental period is $50$, not $100$. The challenge will be that if $100$ fits, any factor will fit as well.

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This problem is actually not new. It is usually called the ACD problem (approximate common divisor problem) or the AGCD problem (approximate greatest common divisor) and there exist several algorithms to solve it.

Although no algorithm is efficient in general, in your case, since the integers are tiny, you can simply try to eliminate the noise and compute the gcd.

Namely, you want to recover $p$ given many values of the form $$x_i = pq_i + r_i$$ where $|r_i|$ is very small, say, smaller than $50$.

Then, you can take two of those values, say, $x_1$ and $x_2$, and compute $d_{a, b} = \gcd(x_1 - a, x_2 - b)$ for all $a, b \in [-50, 50]\cap\mathbb{Z}$. Notice that this step costs only $50^2$ gcds computations, which is very cheap for any computer.

It is guaranteed that one $d_{a,b}$ is equal to $p$. Thus, to check which one is the correct one, just compute the "centered modular reduction" $r_i' := x_i \bmod d_{a,b} \in [-d_{a,b}/2,~ d_{a,b}/2]\cap\mathbb{Z}$ for $i \ge 3$.

If $d_{a,b} = p$, then each $r_i'$ equals $r_i$, therefore, all the values $r_i'$ that you get are small (e.g., smaller than 50). Otherwise, the values $r_i'$ will look randomly distributed in $[-d_{a,b}/2,~ d_{a,b}/2]\cap\mathbb{Z}$.

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