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.