I have this following problem:
$T=nP_1 + mP_2$
With $T$, $P_1$ and $P_2$ real numbers. I have access to those three values, but is it possible to determine $m$ and $n$ ? This looks like the Bézout identity but with real values instead of integers. So would it be a generalization? I also know that $m$ and $n$ are integers between $0$ and $8$.
For context, I am looking at the neighbors of a cell in an array, so 8 surrounding cells with 3 values possible and I want to retrive from the total sum of the values the linear combination of $P_1$ and $P_2$. So I don't have to use several loops and if statements slowing down my program.
Thanks in advance for your answer!
If T, P1 and P2 are totally random, there is generally no solution with n and m integers. Even without the limintation between 0 and 8.
If they are not random, and combinations of 2 well-known values, you have a system with 2 equations and 2 values.
Ex :
$T = 5 \times \sqrt{2}+\sqrt{3}$
$P1 = 6 \times \sqrt{2}+4 \times \sqrt{3}$
$P2 = 7 \times \sqrt{2}-\sqrt{3}$