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j
e
kC1
j
ce
k
for a fixed value of c.
(c) Make a sketch of the functions e
x
and cos.x/ for
1=2
x
1=2. Use this
sketch to argue that
g.x/
D
e
x
cos.x/
(4.192)
has only one zero in this interval. Verify that
g.0/
D
0:
(4.193)
(d) Compute x
1
;:::;x
4
in Newton's method with x
0
D
1=4.Lete
k
D
x
k
x
D
x
k
and compute c
k
as defined by (
4.191
). Conclude, again, that the convergence
seems to be quadratic.
Let us now try to understand the convergence behavior of these cases in a bit
more mathematical way. We consider
f.x/
D
x
2
4;
(4.194)
for which we saw, computationally, in (b) that
j
e
kC1
j
ce
k
:
(4.195)
Our aim is now to derive (
4.195
) analytically.
(e) Show that Newton's method for (
4.194
) can be written in the form
x
k
C
4
2x
k
x
kC1
D
:
(4.196)
(f) Define
x
2
C
4
2x
h.x/
D
(4.197)
and show that
x
2
4
2x
2
h
0
.x/
D
:
(4.198)
(g) Show that
h.x/
2
(4.199)
for any x
2.