Civil Engineering Reference
In-Depth Information
Solving by the quadratic equation, two roots are obtained:
−
()
±−
()
()
−
()
′′
fx
f f(x)
fx
f f(x)
2
n
n
0041
2
′′
()
(()
n
n
∆
x
bbac
a
=
−± −
2
4
2
2
=
=±
()
2
21
2
=± −
()
fx
f f(x)
n
∆
x
′′
()
n
2
With Δ
x = x
n+
1
−
x
n
, Equation 1.4 is as follows:
+
=±−
()
fx
f f(x)
n
x
x
(1.4)
′′
()
n
1
n
n
2
This process is a good tool for finding two roots that are near each other.
This will happen when the slope is close to zero near a root. Double roots
occur when the first derivative is zero, triple roots occur when the first
and second derivatives are zero, and so on. These are shown graphically
in Figure 1.10.
Example 1.13
Newton's second order method
Refine the search from Example 1.7 with a starting value of 1.25 using the
Newton's second order method to increase the accuracy of the approxi-
mate root. For the accuracy test use
e =
0
.
01.
f(x)
f(x)
f(x)
(x)
(x)
(x)
Single root
Double root
Triple root
Figure 1.10.
Newton's second order method.
