Civil Engineering Reference
In-Depth Information
3
2
x
−
84
.
x
+
20 16
.
x
−
13 824
.
=
0
The first and second derivatives of the function must be obtained to find
the slope and curvature at any given value.
()
=− +
fx x
3
84
.
x
2
20 16
.
x
−
13 824
.
′
()
=− +
′′
()
=−
2
fx x
3 68 20 16
6
.
x
.
f x x
16 .
Beginning with
x
n
=
1
.
25, use Equation 1.3 to determine the next value.
(
)
=
(
)
+
2
(
)
−
f
f
125125 84125 016125
.
.
3
−
.
.
.
.
13 824
.
=
0 2041
.
′
(
)
(
)
+
125
.
=
3125
(. )
2
−
16 8125
.
.
20 16
.
=
3 8475
.
′′
(
)
=
(
)
− −
f
1256125 68
.
.
.
9930
.
()
fx
fx
f xfx
fx
0 2401
.
n
x
+
=−
x
=
125
.
−
′′
()()
′
()
(
)
n
1
n
−
930 2041
238475
.
.
′
()
−
n
n
3 8475
.
−
(
)
n
.
2
n
=
1 2001
.
Repeat the process until the desired accuracy is obtained in Table 1.16.
Table 1.16.
Example 1.13 Newton's second order method
x
1.25
1.200143
1.2
f(x)
0.2041
0.00062
0.000000
f ′(x)
3.8475
4.31863
4.320000
f ″(x)
−9.3000
−9.59914
−9.600000
1.12
gRAEffE'S ROOt SQuARing MEtHOD
Graeffe's root squaring method is a root-finding method that was among
the most popular methods for finding roots of polynomials in the 19th
and 20th centuries. This method was developed independently by
