Civil Engineering Reference
In-Depth Information
function of
x
is represented by epsilon,
e
, where |
f
(
x
)| <
e
. Care must
be taken in the selection of the starting point and the increment so that
a root is not missed. This could happen if two roots occur within an
increment and the sign of the function does not change at the successive
values of
x
.
Example 1.8
Refined incremental search method
Refine the search of the function from Example 1.7 between 1.0 and 1.25
using an increment of Δ
x
= 0.25/10 or 1/10th the original increment.
3
2
x
−
84
.
x
+
20 16
.
x
−
13 824
.
=
0
Table 1.11.
Example 1.8 Refined incremental search method
x
1
1.025
1.05
1.075
f(x)
−
1.0640
−
0.9084
−
0.7594
−
0.6170
x
1.1
1.125
1.15
1.175
1.2
f(x)
−
0.4810
−
0.3514
−
0.2281
−
0.1110
0.0000
Since the sign of
f
(
x
) changed between
x
=
1.175 and
x
=
1.2, it is assumed
that a root was passed between those values. The actual root occurs at
x
=
1.2.
1.7
biSEctiOn MEtHOD
After a sign change has occurred in a search method, another way to rap-
idly
converge
(become closer and closer to the same number) on a root is
the bisection method, also known as the half-interval method or the Bol-
zano method developed in 1817 by Bernard Bolzano. This method takes
the bounded increment between two points
x
i
and
x
i
+1
where
f
(
x
i
)
f
(
x
i
+1
) ≤ 0
and divides it in two equal halves or “bisects” the increment. The two sub-
intervals have the first interval from
x
i
to
x
i
+ ½
and the second interval from
x
i
+ ½
to
x
i
+1
as seen in Figure 1.3.
Next, the subinterval containing the root can be found by the follow-
ing algorithm:
f
(
x
i
)
f
(
x
i
+ ½
)
< 0, first interval contains the root
f
(
x
i
)
f
(
x
i
+ ½
)
> 0, second interval contains the root
f
(
x
i
)
f
(
x
i
+ ½
)
=
0,
x
i
+ ½
is the root
