Shape the following problem as a non linear curve fitting problem and write the first iteration step with the gauß newton method.
On a map are n radiostations $S_1,...,S_n$ which coordinates are approximatley known. With the help of a mobile device it´s possible to measure the distances to the stations from your position. With this measurements its possible to calculate your position.
Idea:
My first idea is simple. I create a circle function.
$x_s + r*sin(x)$
$y_s + r*sin(y)$
with $x_s,y_s$ positon of the stations, $x,y \in [0,2*\pi]$ and $r$ the distance. The intersectionpoint of all the circles is your position. But thats not an non linear fitting problem^^. In the next exercise i have to implemnt this problem in matlab and therefore i have three coordiantes: North, East, Height. So it wouldnt work with the circles. Maybe one of you have an idea.
greets
Hint: If $d_i$ is the measured distance from your position $x$ to $S_i$, then $\| x - S_i \|^2 \approx d_i^2 $.
Hint: Avoid taking unnecessary roots ;)
Assume $f$ is the function whose norm you want to minimize. Then, Gauss-Newton is basically solving $$ Df(x)^T Df(x) d = Df(x)^T f(x) $$ and updating $x$ with $$ x \leftarrow x+t d, $$ where $t$ is some step size. For details, you should consult your lecture notes or for example Nocedal's Numerical Optimization.