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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.
 
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