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| Figure 4.38 Numerical Analysis: 2nd Edition |
Part 2 asked us to use 4 satelite positions and time to determine location using algebra and the quatdratic equation. The equations relating the satelite positions and times to our location on Earth can be seen in the figure below.
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| Figure 4.38 Numerical Analysis: 2nd Edition |
These equations begin in terms of x, y, z, and d (the error of our clock on earth). The squared terms
can be removed by subtracting any of the last 3 equations from the first. This yields 3 linear equations.
Using our knowledge of matricies, we can get equations in terms of only x, y, or z, and d.
This can be easily rearanged to solve for x. Once x, y, and z are in terms of d, replacing any of the equations
in figure 4.38 yields a quadratic equation in terms of d. After rearranging the equation to determine the
coefficients for the quadratic, the equation can be solved simply with the quadratic formula. The matlab code
file here performs all of these steps.
The code gave us the following:
d = -.0032
x = -41.7727
y = -16.7892
z = 6370.1