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In-Depth Information
Solving the nonlinear equation f.x/
D
2
C
x
e
x
Tabl e 4. 2
D
0 by Algorithm
4.2
; the number
of iterations k, x
k
and f.x
k
/
k
x
k
f.x
k
/
1
2:209583
4:902331
2
1:605246
1:373837
3
1:259981
0:265373
10
2
4
1:154897
1:880020
10
4
5
1:146248
1:183617
10
9
6
1:146193
4:783945
Example 4.2.
Let
f.x/
D
x
2
2:
We wa n t t o fi n d x
such that
f.x
/
D
0:
One of the two solutions of this problem is given by
p
x
D
2;
and Newton's method is given by
x
k
2
2x
k
x
kC1
D
x
k
;
or equivalently,
x
k
C
2
2x
k
x
kC1
D
;
for k
D
0; 1; 2; : : :.
If we choose x
0
D
1,weget
x
1
D
1:5;
x
2
D
1:41667;
x
3
D
1:41422:
This should be compared with
D
p
x
2
1:41421;
so in just three iterations we get a very accurate approximation.
