Civil Engineering Reference
In-Depth Information
Using Equation 1.1 to solve for a closer point between
x
1
= 1 and
x
2
= 1.25.
()
−
()
()
−
()
=
(
)
−
(
)
xfxxfx
fx fx
102041 1251064
0 2041
.
.
−
.
1
2
2
1
x
=
=
1 2098
.
(
)
3
.
−−−
1 064
.
2
1
Repeat this process until the desired accuracy of the root is achieved as
shown in Table 1.13.
Table 1.13.
Example 1.10 Method of false position
1
2
3
4
1.2018
x
1
1.25
1.2098
interval
1&2
1&3
f(x)
0.2041
0.0417
0.0080
−1.064
1.9
SEcAnt MEtHOD
The secant method is similar to the false position method except that the
two most recent values of
x
(
x
2
and
x
3
) and their corresponding function
values [
f
(
x
2
) and
f
(
x
3
)] are used to obtain a new approximation to the root
instead of checking values that bound the root. This eliminates the need to
check which subinterval contains the root. The variable renaming process
for iteration is as follows:
x
1
=
x
2
and
x
2
=
x
3
f
(
x
1
) =
f
(
x
2
) and
f
(
x
2
)
= f
(
x
3
)
In some instances
interpolation
occurs, this is when the new value is
between the previous two values. In others,
extrapolation
occurs, meaning
the new value is not between the previous two values. Interpolation was
shown in Figure 1.4 and extrapolation is shown in Figure 1.5.
Example 1.11
Secant method
Refine the search of the function from Example 1.7 between 1.0 and 1.25
using the secant method to increase the accuracy of the approximate root.
For the accuracy test use
e
= 0.01.
3
2
x
−
84
.
x
+
20 16
.
x
−
13 824
.
=
0
