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$($The LORAN $($LOng RAnge Navigation$)$ system calculates the position of a boat at sea using signals from fixed transmitters. From the time differences of the incoming signals, the boat obtains differences of distances to the transmitters. This leads to two equations each representing hyperbolas defined by the differences of distance of two points $($foci$).$ An example of such equations from are$).$

$f_1(x,y)=\frac{y^2}{300-{186}^2}+\frac{x^2}{{186}^2}=1$

$f_2(x,y)=\frac{(y-500{)}^2}{{279}^2}-\frac{(x-300{)}^2}{({500}^2-279{)}^2}=1$

Solving two quadratic equations with two unknowns, would require solving a $4$ degree polynomial equation. We could do this by hand, but for a navigational system to work well, it must do the calculations automatically and numerically. We note that the Global Positioning System $($GPS$)$ works on similar principles and must do similar computations.

Solve this set of nonlinear equation by hand $($Newton$\text{'}$s Method$)$:

$f_1(x,y)=y+1\mathrm{.}02x^3$

$f_2(x,y)=-0\mathrm{.}9x+y^3$

Graph the solution. Hence, show that $(1,0),$ is an initial point of solution.

Solve the above problem by hand using NEWTON'S METHOD $($ALSO KNOWN AS NEWTON, RAPHSON METHOD$).$

Write a computer algorithm $-$ code to solve the above NEWTON'S METHOD $($ALSO KNOWN AS NEWTON, RAPHSON METHOD$).$

Show the code, hence verify your computer algorithm and the hand results.