Civil Engineering Reference
In-Depth Information
Repeat the process until the desired accuracy is obtained in Table 1.15.
Table 1.15.
Example 1.12 Newton-Raphson method
x
1.25
1.1969
1.19999
f(x)
0.2041
−0.0132
−0.00004
f
′
(x)
3.8475
4.3493
4.32010
1.11
nEWtOn'S SEcOnD ORDER MEtHOD
Newton's second order method is often a preferred method to determine
the value of a root due to its rapid convergence and extremely close
approximation. This method also includes the second derivative of the
function or the curvature to find the approximate root. The equation
f
(
x
) = 0 is considered as the target for the root. Figure 1.9 shows the plot
of the actual function.
The Taylor series expansion was discovered by James Gregory and
introduced by Brook Taylor in 1715 (Taylor 1715). The following is a
Taylor series expansion of
f
(
x
) about
x
=
x
n
:
′′
()
()
+
′′′
()
()
+…
2
3
f x x
∆
f
x
∆
x
(
)
=
()
+ ′
()
(
+
n
n
fx fx fx x
∆
n
+
1
n
n
2
!
3
!
For a means of determining a value of Δ
x
that will make the Taylor series
expansion go to zero, the first three terms of the right hand side of the
equation are set equal to zero to obtain an approximate value.
f(x)
∆x
f(x)
x
n+1
(x)
x
n
Figure 1.9.
Newton's second order method.
