lecture13 - Lecture 13 Nonlinear Systems Newtons Method An...

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Lecture 13 Nonlinear Systems - Newton’s Method An Example 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 dif- ferences 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 [2] are: x 2 186 2 - y 2 300 2 - 186 2 = 1 and ( y - 500) 2 279 2 - ( x - 300) 2 500 2 - 279 2 = 1 . (13.1) 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. Vector Notation In general, we can usually find solutions to a system of equations when the number of unknowns matches the number of equations. Thus, we wish to find solutions to systems that have the form: f 1 ( x 1 ,x 2 ,x 3 ,...,x n ) = 0 f 2 ( x 1 ,x 2 ,x 3 ,...,x n ) = 0 f 3 ( x 1 ,x 2 ,x 3 ,...,x n ) = 0 . . . f n ( x 1 ,x 2 ,x 3 ,...,x n ) = 0 . (13.2) For convenience we can think of ( x 1 ,x 2 ,x 3 ,...,x n ) as a vector x and ( f 1 ,f 2 ,...,f n ) as a vector- valued function f . With this notation, we can write the system of equations (13.2) simply as: f ( x ) = 0 , i.e. we wish to find a vector that makes the vector function equal to the zero vector.
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