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x
3
x
2
x
1
x
0
f
(
x
)
Fig. 4.5
A graphical illustration of Newton's method
we get
f.x
1
/
f
0
.x
1
/
:
x
2
D
x
1
(4.62)
More generally, we have
f.x
k
/
f
0
.x
k
/
:
x
kC1
D
x
k
(4.63)
This process is illustrated in Fig.
4.5
.
We will see in the next section that this derivation of Newton's method motivates
the secant method, for which we do not need an explicit evaluation of f
0
.x/.
4.4
The Secant Method
We noted above that if we have a linear model F
D
F.x/ of f
D
f.x/, we can
solve F.x/
D
0 in order to find the approximation of the solution of f.x/
D
0.The
secant method is based on the same idea, but instead of using the Taylor series, we
use interpolation based on a linear function. Suppose we have two values x
0
and x
1
close to x
,where
f.x
/
D
0:
(4.64)
