Civil Engineering Reference
In-Depth Information
f(x)
f(x)
f(x
i
)
f(x
i+1/2
)
(x)
x
i
x
i+1/2
x
i+1
x
f(x
i+1
)
∆x
∆x
Figure 1.3.
Bisection method.
The process is continued by bisecting the subinterval containing the root
and repeating the procedure until the desired accuracy is achieved. After
n
bisections, the size of the original interval has been reduced by a factor
of 2
n
.
Example 1.9 Bisection method
Refine the search of the function from Example 1.7 between 1.0 and 1.25
using the bisection method to increase the accuracy. Use
e
= 0.01 that is
|
f
(
x
)| <
e
:
3
2
x
−
84
.
x
+
20 16
.
x
−
13 824
.
=
0
Begin by solving the equation for 1 and 1.25, which was done in
Example 1.7. The sign changes, so a root lies between the two. We also
know from the refined incremental search method the root should fall
between 1.175 and 1.2. Next, bisect the increment between 1 and 1.25,
which is a value of 1.125. Evaluate the function at that point and com-
pare the two subintervals for the sign changes. Also, check to see if
the desired accuracy on
f
(
x
) is achieved. This occurs between 1.125
and 1.25, so that interval is subdivided again at 1.1875. Continue the
bisections until the desired accuracy is achieved. Note in Table 1.12
this occurs at
x
= 1.1992 where
f
(
x
) = -0.0034. This is the last bisection
between 1.1953 and 1.2031.
