Homework 4 due Thursday November 4 before 13:30 in corre
sponding group folder on door of oﬃce CM110
•
Prove that the sequence generated by the iterative formula
p
n
+1
= 1

sin
p
n
converges
to the real root of the equation sin
x
+
x

1 = 0 for every real starting value
p
0
. Use
this formula, starting with
p
0
= 0
.
5, and Aitken’s acceleration method to find that
root correct to three decimal places.
The iterative method is
p
n
+1
=
g
(
p
n
) where
g
(
x
) = 1

sin
x
. Hence a fixed point,
p
=
g
(
p
) satisfies
p
= 1

sin
p
⇐⇒
f
(
p
) := sin
p
+
p

1 = 0
(0.1)
i.e.
p
is a root of the equation
f
(
p
) = 0. For any starting value
p
0
∈
(
∞
,
∞
),
sin
p
0
∈
[

1
,
1] =
⇒
p
1
=
g
(
p
0
) = 1

sin
p
0
∈
[0
,
2]
so we need only restrict our attention to a fixed point in the interval [0
,
2].
–
We show directly that equation (0.1) has a unique solution
p
= 1

sin
p
∈
[0
,
2].
Let
f
(
x
) = sin
x
+
x

1, then
f
0
(
x
) = cos
x
+ 1
>
0, for all
x
∈
[0
,
2]. Moreover
we note that
f
(0) =

1
<
0 and
f
(2) = 1 + sin 2
>
0.
We deduce that
f
(
·
)
is a strictly increasing continuous function which changes sign, hence from the
Intermediate Value theorem there exists a unique
p
∈
[0
,
2] such that
f
(
p
) = 0.
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 Fall '11
 Dr.A.Wacher
 Numerical Analysis, Sin, Order theory, Monotonic function, Convex function, oﬃce CM110

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