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Homework 5 due Thursday November 11 before 13:30 in corre
sponding group folder on door of oﬃce CM110
1. (a) The equation
x
3
+
x

1 = 0 has one real root near to 0.7. Discuss brieﬂy,
and without actually implementing the iterative method, the convergence of
iterative methods based on the following rearrangements of the equation, for
starting values suﬃciently close to the root.
x
= 1

x
3
,
x
= 1
/
(1 +
x
2
)
,
x
= (1

x
)
1
/
3
(b) If
f
(
x
) = (
x

p
)
α
h
(
x
) for some integer
α >
1, and
h
(
p
)
6
= 0, show that the
NewtonRaphson formula gives a sequence which is linearly convergent to
p
.
Show that in this case the formula
p
n
+1
=
p
n

αf
(
p
n
)
/f
0
(
p
n
)
generates a sequence which, for suitable starting values, has quadratic conver
gence.
——————————————————————————————
(a)
i. Let
g
(
x
) := 1

x
3
. Since
g
0
(
x
) =

3
x
2
it follows for
g
0
(0
.
7)
<

1 and by the
divergence theorem we expect this iteration will not converge.
ii. Let
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 Fall '11
 Dr.A.Wacher
 Numerical Analysis

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