4.10 Complex Networks
137
Fig. 4.14 Second-degree
polynomial with two roots
(a, 0) and (b, 0)
a
b
f(x)
x
y
x n = x n−1 −
f (x n−1 )
f (x n−1 )
(4.26)
That is, the next estimate for f (x) = 0 is obtained taking the difference between
the value of a seed point (x n − 1) and the ratio of the value of f (x) at that seed point
and its derivative at that same point. For the series to converge, the starting point,
x n=1 , must belong to the existing field of the function.
It should be noted that the fact that the method finds a solution does not necessarily
indicate that it is a physically meaningful solution to any problem under investigation
since the value found depends on the type of function under analysis and the seed
values used. For example, taking the function shown in Fig. 4.14, if Newton’s method
is used with a starting value (0 < x n=1 > a) it will locate the root (a, 0); while if it
is used with a starting value belonging to the interval (a < x n=1 > b), depending on
whether this value occurs before the minimum value or after, it could converge on
either root (a, 0) or (b, 0).
This highlights that, when solving real-life problems, there is a technical necessity
to identifying points close to the solution or having a system for checking solutions
either graphically or by some other independent numerical method.
Finally, it must be remembered that Newton’s method cannot be applied if f
(x)
= 0, since the tangent will be horizontal and therefore the function will not cut the
x-axis.
Note: The choice of the function to be optimized must fulfill the following
conditions:
1. Be related to the process to be studied.
2. Include all known variables together with those to be determined and reflect the
relationship between them.
3. Include as few as possible unknown variables.
In addition:
4. The function should be as simple as possible to facilitate calculations, for
example, Z = X
2
+ Y
2 . If functions of the type Z = X Y are considered, the
solution may prove difficult to find as sign changes in one of the terms will cause
137
Fig. 4.14 Second-degree
polynomial with two roots
(a, 0) and (b, 0)
a
b
f(x)
x
y
x n = x n−1 −
f (x n−1 )
f (x n−1 )
(4.26)
That is, the next estimate for f (x) = 0 is obtained taking the difference between
the value of a seed point (x n − 1) and the ratio of the value of f (x) at that seed point
and its derivative at that same point. For the series to converge, the starting point,
x n=1 , must belong to the existing field of the function.
It should be noted that the fact that the method finds a solution does not necessarily
indicate that it is a physically meaningful solution to any problem under investigation
since the value found depends on the type of function under analysis and the seed
values used. For example, taking the function shown in Fig. 4.14, if Newton’s method
is used with a starting value (0 < x n=1 > a) it will locate the root (a, 0); while if it
is used with a starting value belonging to the interval (a < x n=1 > b), depending on
whether this value occurs before the minimum value or after, it could converge on
either root (a, 0) or (b, 0).
This highlights that, when solving real-life problems, there is a technical necessity
to identifying points close to the solution or having a system for checking solutions
either graphically or by some other independent numerical method.
Finally, it must be remembered that Newton’s method cannot be applied if f
(x)
= 0, since the tangent will be horizontal and therefore the function will not cut the
x-axis.
Note: The choice of the function to be optimized must fulfill the following
conditions:
1. Be related to the process to be studied.
2. Include all known variables together with those to be determined and reflect the
relationship between them.
3. Include as few as possible unknown variables.
In addition:
4. The function should be as simple as possible to facilitate calculations, for
example, Z = X
2
+ Y
2 . If functions of the type Z = X Y are considered, the
solution may prove difficult to find as sign changes in one of the terms will cause