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Here is a system of 3 equations with 3 unknowns $a$, $b$ and $c$.

$\begin{cases} a^2+\sigma \cdot a\cdot b + \sigma \cdot a \cdot c = n_1 \\ b^2+\sigma \cdot a\cdot b + \sigma \cdot b \cdot c = n_2 \\ c^2+\sigma \cdot a\cdot c + \sigma \cdot b \cdot c = n_3, \end{cases}$

where $0<\sigma<1$, $n_1,n_2,n_3>0$ are all known.

This system seems quite simple but I cannot think of any method to solve it.

Q1: Is there a method to check the condition of this system to have real solution for $a,b,c$?

Q2: Is there a method to solve it analytically? Or at least to approximate it numerically?

Anthony
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1 Answers1

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This is a very difficult problem and do not expect any analytical solution.

What I would do is to consider the three equations

$$f_1= a^2+\sigma \, a\, b + \sigma \, a \, c - n_1 $$ $$f_2= b^2+\sigma \, a\, b + \sigma \, b \, c - n_2 $$ $$f_3= c^2+\sigma \, a\, c + \sigma \, b \, c - n_3 $$ and define, as a norm, $$\Phi(a,b,c)=f_1^2+f_2^2+f_3^2$$ and try to minimize it. If, at the solution, $\Phi=0$, you will have one solution.

For a test, I used $\sigma=0.678$, $n_1=18$, $n_2=28$, $n_3=39$ and the optimizer worked like a charm, leading to $\Phi=1.34\times 10^{-29}$ and $$a=2.31796 \qquad \qquad b=3.44477\qquad \qquad c=4.58986$$ The initial data were generated using $a=2.34$, $b=3.45$, $c=4.56$ and the computed $n_i$ were rounded to the next integer.

In a second run, I added $15$, $10$ and $5$ to the $n_i$; no problem.

For me, this would be the safest approach.

For sure, it would be better to provide initial estimates for $(a,b,c)$. Since there are only three parameters, a preliminary grid search could be a good idea.

Claude Leibovici
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