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LECTURE 2.3
Dr. Vyas
Overview: (Covers a part of section 2.4)
•
NewtonRaphson Method
•
Secant Method
•
Examples/Demos
NEWTONRAPHSON METHOD:
A. Algebraic Description
1.
Theorem 2.5
: Assume that
2
[ , ]
f
C a b
∈
(
meaning function is continuous upto its
second derivative in the given interval
) and that there exists a number
[ , ]
p
a b
∈
such that
( )
0
f p
=
. If
( )
0
f
p
′
≠
, then there exists a
0
δ
such that the sequence
{ }
0
k
k
p
∞
=
defined by the iteration
1
1
1
1
(
)
(
)
(
)
k
k
k
k
k
f p
p
g p
p
f
p




=
=

′
for
1, 2,
k
=
K
where,
( )
( )
( )
f x
g x
x
f x
≡

′
will converge to
p
for any initial approximation
0
[
,
]
p
p
p
∈

+
.
2.
The function
( )
g x
is called the
NewtonRaphson iteration function
.
3.
As can be observed the recursive relation
1
(
)
k
k
p
g p

=
is an indicator of the
application of the
fixed point iteration method
. Therefore, one can also think of
this process as that of finding a fixed point of the function
( )
g x
(Not
( )
f x
!)
4.
Recall, the Taylor expansion of a function about a point
0
x
x
=
may be expressed
as
( )
(
1)
1
0
0
0
0
( )
( )
( )
(
)
(
)
!
(
1)!
k
n
n
k
n
k
f
x
f
c
f x
x
x
x
x
k
n
+
+
=
=

+

+
∑
. The assumptions that
underlies this expression are that,
1
0
0
[ , ],
[ , ],
( )
( ,
)
n
f
C
a b x
a b c
c x
x x
+
∈
∈
=
∈
5.
Let
0
0
x
p
=
be the starting point of the iterative process. Therefore, the Taylor
expansion (approximated to
two
term expansion plus remainder) can be rewritten
as,
(2)
(1)
2
0
0
0
0
( )
( )
(
) (
)
(
)
(
)
2!
f
c
f x
f p
x
p
f
p
x
p
=
+

+

. The superscript numbers
are the order of the derivatives of the function. The primed notation will be
followed henceforth to stay with the textbook nomenclature.
6.
If
p
is the fixed point of the function then,
( )
( )
0
g p
p
f p
=
⇒
=
. Therefore,
using this information in the Taylor expansion,
2
0
0
0
0
( )
0
( )
(
) (
)
(
)
(
)
2!
f
c
f p
f p
p
p
f
p
p
p
′′
′
=
=
+

+

.
7.
Further, if the starting point
0
p
is close enough (practically, sufficiently less than
unity) to the fixed point
p
, then the last term in the above expression can be
neglected. As a result, the following
approximation
results,
0
0
0
0
0
0
(
)
(
) (
)
(
)
0
(
)
f p
f p
p
p
f
p
p
p
f
p
′
+

≈
⇒
≈

′
. Geometrically, we are
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approximating the curve by its tangent at a point. This is valid in so far one is
close to the point of tangency.
B. Geometric Description
8.
The slope of tangent line between points
0
0
(
, (
))
p
f p
and
1
(
,0)
p
is given by
0
1
0
0
(
)
f p
slope
p
p

=

. However, recall that for a function
( )
y
f x
=
, the slope is also
given by its first derivative
( )
y
f x
′
=
. The first derivative evaluated at the point
of tangency gives the slope of the tangent line. Therefore, the slope of the tangent
line is
0
0
0
1
0
1
0
0
0
(
)
(
)
(
)
(
)
f p
f p
f
p
p
p
p
p
f
p

′
=
⇒
=

′

9.
This approximation can be used to refine until fixed point (or root) is located.
Therefore, the first approximation to the root is,
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This note was uploaded on 04/14/2008 for the course MATH 353 taught by Professor Vyas during the Spring '08 term at University of Delaware.
 Spring '08
 Vyas
 Math, Algebra

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