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An Alternative Derivation
Let us first briefly recall the derivation of Newton's method. We assume that x
0
is a
given value close to x
,wherex
satisfies
f.x
/
D
0:
(4.48)
By a Taylor expansion, we have
f.x
0
C
h/
f.x
0
/
C
hf
0
.x
0
/:
(4.49)
We note that
f.x
1
/
D
f.x
0
C
h/
0
(4.50)
can be obtained by choosing
h
D
f.x
0
/
f
0
.x
0
/
:
(4.51)
This motivates the choice
f.x
0
/
f
0
.x
0
/
;
x
1
D
x
0
(4.52)
and more generally we have
f.x
k
/
f
0
.x
k
/
:
x
kC1
D
x
k
(4.53)
The key observation in this derivation is to utilize the Taylor series (
4.49
).
We can also use the Taylor series in a somewhat different manner. Recall from
(1.39) on page 20 that we have the Taylor series of f in the form
f.x/
D
f.x
0
/
C
.x
x
0
/f
0
.x
0
/
C
..x
x
0
/
2
/:
(4.54)
O
Let us now define the linear model
F
0
.x/
D
f.x
0
/
C
.x
x
0
/f
0
.x
0
/
(4.55)
of f around x
D
x
0
.InFig.
4.4
we see that F
0
approximates f around x
D
x
0
.
What are the properties of F
0
D
F
0
.x/? We first note that
F
0
.x
0
/
D
f.x
0
/:
(4.56)
Second, we observe that
