Civil Engineering Reference
In-Depth Information
f(x)
f(x)
f(x
n
)
x
n+1
(x)
x
n
f′(x
n
)
Figure 1.6.
Newton-Raphson method or Newton's tangent.
1669 (Newton 1669). Once an approximate root
x
n
has been found, not
only is the function,
f
(
x
n
), used, but the slope of the function at that point,
f ′
(
x
n
), is also incorporated to converge to the root more rapidly. The slope
of the function is found from the first derivative of the function evaluated
at a point. This only requires the use of one value to be known. The slope
intersects the
x
-axis at a value
x
n+
1
as shown in Figure 1.6 and the relation-
ship is given in Equation 1.2.
′
()
=
()
−
∴=−
()
fx
xx
x
fx
fx
fx
n
x
n
(1.2)
′
()
n
n
+
1
n
n
n
+
1
n
Repeat the process using a new value until convergence occurs. Conver-
gence may not occur in the following two cases:
•
f ′′
(
x
n
), (curvature) changes sign near a root, shown in Figure 1.7.
• Initial approximation is not sufficiently close to the true root and
the slope at that point has a small value, shown in Figure 1.8.
Example 1.12 Newton-Raphson method
Refine the search from Example 1.7 with a starting value of 1.25 using
the Newton-Raphson method to increase the accuracy of the approximate
root. For the accuracy test use
e
= 0.01.
x
3
−
84
.
x
2
+
20 16
.
x
−
13 824
.
=
0
