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I have a system of equations like this

$$ \sum_i^n \frac{1}{x+a_i y+b_iz}=S_1,\\ \sum_i^n \frac{a_i}{x+a_i y+b_iz}=S_2,\\ \sum_i^n \frac{b_i}{x+a_i y+b_iz}=S_3, $$ where $n,a_i,b_i,S_{1,2,3}$ are parameters (known), and I would like to solve $(x,y,z)$. Here $n$ is a big number, such as 100.

Is there any analytical or numerical method to solve this kind equation? Or I used some stupid numerical method and got a solution, is there any method to prove that this equation can have and only have one solution?

Qi Zhong
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  • Sometimes there are no solutions, sometimes there are infinitely many solutions. For example, if all $a_i=a$, all $b_i=b$, then the equations become $x+ay+bz=n/S_1=na/S_2=nb/S_3$. If $n/S_1=na/S_2=nb/S_3$ we have infinitely many solutions. If $n/S_1=na/S_2=nb/S_3$ is not satisfied, then there are no solutions. – plop May 13 '21 at 21:00
  • @plop Thanks for your reply. Since I use this equations for a real physical system, I know that there must be a solution. Just not sure if there is a standard method to solve this kind equations. – Qi Zhong May 14 '21 at 02:17

1 Answers1

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I would not expect a formal solution for this problem.

But, using stupid numerical methods

  • Newton-Raphson would a good idea (provided you have "reasonable" estimates) - very simple derivatives
  • What I would prefer is minimization of the norm $$\Phi(x,y,z)=\Bigg[\sum_{i=1}^n \frac{1}{x+a_i y+b_iz}-S_1\Bigg]^2+$$ $$\Bigg[\sum_{i=1}^n \frac{a_i}{x+a_i y+b_iz}-S_2\Bigg]^2+\Bigg[\sum_{i=1}^n \frac{b_i}{x+a_i y+b_iz}-S_3\Bigg]^2$$ and you can easily provide the analytical Jacobian and Hessian.
Claude Leibovici
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