Civil Engineering Reference
In-Depth Information
Table 1.12.
Example 1.9 Bisection method
1
2
3
4
x
1
1.25
1.125
1.1875
interval
1&2
2&3
f(x)
−1.0640
0.2041
−0.3514
−0.0548
5
6
7
x
1.2188
1.2031
1.1953
1.1992
interval
2&4
4&5
4&6
6&7
f(x)
0.0793
0.0135
−0.0204
−
0.0034
1.8
MEtHOD Of fALSE POSitiOn OR LinEAR
intERPOLAtiOn
Although the bisection method can be used to reach convergence, other
methods such as false position provide the same accuracy more rapidly.
The process is similar to the bisection method in that between
x
i
and
x
i
+1
where
f
(
x
i
)
f
(
x
i
+1
) ≤ 0 a root exists. Refer to
x
i
and
x
i
+1
as
x
1
and
x
2
, respec-
tively. A straight line connecting
x
1
and
x
2
intersects the x-axis at a new
value, say
x
3
, which is closer to the root than either
x
1
or
x
2
. Thus, by sim-
ilar triangles, the value of
x
3
can be found.
()
−
()
−
−
()
fx fx
xx
0
fx
xx
xxfx
xx
−
()
−
()
−
∴=−
()
2
1
1
=
2
1
3
1
1
fx fx
2
1
3
1
2
1
This equation can also be rewritten as follows:
()
−
()
()
−
()
xfxxfx
fx fx
1
2
2
1
x
=
(1.1)
3
2
1
The relationship between
x
1
,
x
2
, and
x
3
can be seen in Figure 1.4.
f
(
x
1
)
f
(
x
3
)
< 0, first interval contains the root
f
(
x
1
)
f
(
x
3
) > 0, second interval contains the root
f
(
x
1
)
f
(
x
3
) =
0,
x
3
is the root
