This preview shows pages 1–4. Sign up to view the full content.
Chapter 03.03
Bisection Method of Solving a Nonlinear Equation
After reading this chapter, you should be able to:
1.
follow the algorithm of the bisection method of solving a nonlinear equation,
2.
use the bisection method to solve examples of finding roots of a nonlinear equation,
and
3.
enumerate the advantages and disadvantages of the bisection method.
What is the bisection method and what is it based on?
One of the first numerical methods developed to find the root of a nonlinear equation
0
)
(
=
x
f
was the bisection method (also called
binarysearch
method).
The method is based
on the following theorem.
Theorem
An equation
0
)
(
=
x
f
, where
)
(
x
f
is a real continuous function, has at least one root
between
x
and
u
x
if
0
)
(
)
(
<
u
x
f
x
f
(See Figure 1).
Note that if
0
)
(
)
(
u
x
f
x
f
, there may or may not be any root between
x
and
u
x
(Figures 2 and 3).
If
0
)
(
)
(
<
u
x
f
x
f
, then there may be more than one root between
x
and
u
x
(Figure 4).
So the theorem only guarantees one root between
x
and
u
x
.
Bisection method
Since the method is based on finding the root between two points, the method falls
under the category of bracketing methods.
Since the root is bracketed between two points,
x
and
u
x
, one can find the mid
point,
m
x
between
x
and
u
x
.
This gives us two new intervals
1.
x
and
m
x
, and
2.
m
x
and
u
x
.
03.03.1
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document 03.03.2
Chapter 03.03
Figure 1
At least one root exists between the two points if the function is real, continuous,
and changes sign.
Figure 2
If the function
)
(
x
f
does not change sign between the two points, roots of the equation
0
)
(
=
x
f
may still exist between the two points.
f
(
x
)
x
ℓ
x
u
x
f
(
x
)
x
ℓ
x
u
x
Bisection Method
03.03.3
Figure 3
If the function
)
(
x
f
does not change sign between two points, there may not be
any roots for the equation
0
)
(
=
x
f
between the two points.
Figure 4
If the function
)
(
x
f
changes sign between the two points, more than one root for
the equation
0
)
(
=
x
f
may exist between the two points.
Is the root now between
x
and
m
x
or between
m
x
and
u
x
?
Well, one can find the sign of
)
(
)
(
m
x
f
x
f
, and if
0
)
(
)
(
<
m
x
f
x
f
then the new bracket is between
x
and
m
x
, otherwise,
it is between
m
x
and
u
x
.
So, you can see that you are literally halving the interval.
As one
repeats this process, the width of the interval
[
]
u
x
x
,
becomes smaller and smaller, and you
can zero in to the root of the equation
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
This is the end of the preview. Sign up
to
access the rest of the document.
This note was uploaded on 05/01/2011 for the course CHBE 2120 taught by Professor Gallivan during the Spring '07 term at Georgia Institute of Technology.
 Spring '07
 Gallivan

Click to edit the document details