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ME218 Engineering Computational Methods
Lecture 3 Notes
Methods for Solving Nonlinear Equations (Continued)
Newton Raphson’s Method
This technique is a rootfinding algorithm that uses the first few terms of the Taylor
series of a function
f(x)
in the vicinity of a suspected root,
x
0
.
....
)
(
2
)
(
)
(
)
(
)
(
)
(
0
2
0
0
0
0
!
"
"
#
!
"
#
!
$
%
x
f
x
x
x
f
x
x
x
f
x
f
higher terms
Now, for x to be the root,
f(x)
= 0.
Thus, taking only the linear terms in the Taylor series expansion,
)
(
)
(
)
(
0
)
(
0
0
0
x
f
x
x
x
f
x
f
"
#
!
$
$
Rearranging,
x = x
0
– f(x
0
)/f’(x
0
)
, where
x
0
is the initial guess.
Generalizing the iterative process:
x
N+1
=
x
N
–
f(x
N
)
/
f’(x
N
)
, till 
x
N+1
–
x
N
 < e
This method converges faster than the Bisection and the RegulaFalsi methods, but
suffers from the drawback that the process might hit a local extremum and make the
derivative term go to zero. In that case, the method becomes unstable, as there is a
division by 0!
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2
Secant Method
To overcome the need in the Newton‐Raphson iterative scheme to evaluate first
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 Fall '08
 Unknown
 Calculus, Derivative, Taylor Series, Rootfinding algorithm

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