Civil Engineering Reference
In-Depth Information
equations. A search is performed over a given range of values for
x
usually
denoted as minimum and maximum values,
x
min
and
x
max
. An increment on
x
of Δ
x
is used to determine the successive values of
f
(
x
)
.
Each consecutive
pair of functions of
x
are compared and when their signs are different a
root of
x
has been bounded by the two values of
x
. Written in algorithmic
form, a sign change occurs between
x
i
and
x
i
+1
if
f
(
x
i
)
f
(
x
i
+1
)
≤
0. The sign
change generally indicates a root has been passed but could also indicate
a discontinuity in the function. This process is illustrated graphically in
Figure 1.1.
Example 1.7
Incremental search method
Determine the first approximate root of the following function starting at
x
min
= 0 and using an increment Δ
x
= 0.25:
x
3
−
84
.
x
2
+
20 16
.
x
−
13 824
.
=
0
Table 1.10.
Example 1.7 Incremental search method
x
0
0.25
0.5
0.75
1
1.25
f(x)
−
13.8240
−
9.2934
−
5.7190
−
3.0071
−
1.0640
0.2041
f(x)
f(x)
f(x
i
)
(x)
x
i
x
x
i+1
f(x
i+1
)
∆x
∆
x
Figure 1.1.
Incremental search method.
