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I feel something like this must have been asked before, but can't really find it. Context below.

$\sum_{j=0}^{M}\left[(e^{-\sqrt{(x-x_{j})^2+(y-y_{j})^2+(z-z_{j})^2}}-e^{-r_{j}}) \frac{-(x-x_{j})}{\sqrt{(x-x_{j})^2+(y-y_{j})^2+(z-z_{j})^2}} \right]=0$

$\sum_{j=0}^{M}\left[(e^{-\sqrt{(x-x_{j})^2+(y-y_{j})^2+(z-z_{j})^2}}-e^{-r_{j}}) \frac{-(y-y_{j})}{\sqrt{(x-x_{j})^2+(y-y_{j})^2+(z-z_{j})^2}} \right]=0$

$\sum_{j=0}^{M}\left[(e^{-\sqrt{(x-x_{j})^2+(y-y_{j})^2+(z-z_{j})^2}}-e^{-r_{j}}) \frac{-(z-z_{j})}{\sqrt{(x-x_{j})^2+(y-y_{j})^2+(z-z_{j})^2}} \right]=0$

where $x_{j}$, $y_{j}$ , $z_{j}$ and $r_{j}$ are constants for $j=0$ to $j=M$

CONTEXT

I have point $(x,y,z)$ and a set of points $(x_{j},y_{j},z_{j})$ for $j=0$ to $j=M$. That set of points move on space almost independently of each other, and I want to find the coordinates of a new point $(x,y,z)$ that mimicks the distances to each point $(x_{j},y_{j},z_{j})$ with the minimum possible error. Also, if a point $(x_{j},y_{j},z_{j})$ is originally closer to $(x,y,z)$, the error with respect to this point should be lower.

For this, I have defined the following function:

$F=\sum_{j=0}^{M}\left[(e^{-r_{j}}-e^{-r_{j_{0}}})^{2}\right]$

where $r_{j}$ are constants and $r_{j}=\sqrt{(x-x_{j})^{2}+(y-y_{j})^{2}+(z-z_{j})^{2}}$

To find the minimum values of x, y and z, I calculated the partial derivatives of F with respect to x, y and z and ended up with the displayed system of equations.

Thanks

user3091644
  • 103
  • Welcome to MSE. Your question is phrased as an isolated problem, without any further information or context. This does not match many users' quality standards, so it may attract downvotes, or closed. To prevent that, please [edit] the question. This will help you recognize and resolve the issues. Concretely: please provide context, and include your work and thoughts on the problem. These changes can help in formulating more appropriate answers. – José Carlos Santos Apr 11 '22 at 08:41
  • Let's consider most simple non-trivial case. $M=1$, 1D. Then equation becomes $e^{-|x-x_0|}+e^{-|x-x_1|}=e^{-r_0}+e^{-r_1}$. This equation can be solved analytically. Increasing $M$ does not change this fact, but increases number of cases substantially, so the answer will severely depend on relations between $x_i$ and $r_i$. I don't understand what can be profit from such answer as it cannot be use for calculations. Increasing dimensionality of problem makes this problem more complex, and I believe there is no reason to solve it in general form. – Ivan Kaznacheyeu Apr 11 '22 at 09:35

0 Answers0