3-1
CHAPTER 3
ROOT FINDING
This chapter is about different root searching techniques.
Given a function
f
(
x
), we are interested to look for its root
ξ
(i.e.
f
(
ξ
) = 0). The general idea is to revise the value of
x
through
a
repeated
process.
The
goal
is
for
x
to
asymptotically approach
ξ
as the number of iterations
increases. Four iteration methods are considered:
bisection
,
fixed-point iteration
,
Newton’s method
, and
secant method
.
Their basic concepts, algorithms, and implementations in
Matlab are presented.
In order to illustrate the relevancy of root finding to electrical
engineering, we consider a diode circuit problem throughout
this chapter.
V
0
R
i
voltage
x
The diode is governed by the diode equation:
)
1
(
/
−
=
th
V
x
S
e
I
i
where
I
S
and
V
th
are known parameters of the device. We are
interested to find the voltage
x
across the diode. Since
EE 3108 Semster B 2007/2008
S C Chan
3-2
iR
x
V
+
=
0
,
we have (by combining the two equations):
)
1
(
/
0
−
=
−
th
V
x
S
e
R
I
x
V
0
2
10
5
026
.
0
/
15
=
−
+
×
∴
−
x
e
x
------(*)
Thus, the problem is a root finding problem of the function
2
10
5
)
(
026
.
0
/
15
−
+
×
=
−
x
e
x
f
x
.
The function is plotted below.
0.8
0.81
0.82
0.83
0.84
0.85
0.86
0.87
0.88
0.89
0.9
−
2
−
1
0
1
2
3
4
5
We can observe from the graph that the root is
8596
.
0
=
ξ
(4
s.f.). However, we need better methods because it is not
effective to plot a graph for every problem.

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