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Numerical Analysis II
Dr Abigail Wacher
Oct 28, 2010
1
Newton's Method
gx
=
x
K
fx
f
'
Since
g
'
=
''
'
2
'
p
= 0 as long as
'
s
0 so care must be taken if
has a double root or higher multiplicity root.
Provided the starting value is "close" to
and
'
s
0, speed of convergence is quadratic.
Motivating using the definition of the derviative of a function, replace
'
by the secant.
'
n
z
K
K
1
K
K
1
As the points get closer together secants method approaches the tangent
Secant Method
C
1
=
K
K
K
1
K
C
1
Note resemblance to Regula Falsi!
The secant method is slower than the Newton method, but is still superlinear.
It can be shown that for the secant method
e
C
1
z
ce
k
K
1
where
c
is a constant
=
K
From this one can derive
C
1
z
h
where
h
=
and the exponent
satisfies
2
K
K
1 = 0 taking the positive root
=
1
2
1
C
5
z
1.61803398875
This number is related to the "Golden Section" popular in the ancient Greek civilization.
There is also relation with the Fibonacci Sequence.
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This note was uploaded on 02/09/2011 for the course MATH 2051 taught by Professor Dr.a.wacher during the Fall '11 term at Durham.
 Fall '11
 Dr.A.Wacher
 Numerical Analysis

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