MATH 2070U Midterm
March 1st, 2018
Page 7 of 11
4. [7 marks] Let the matrix
A
be given by
A
=
1
2
0
3
1
1
2
1

1
.
Compute the
LU
factorisation
PA
=
LU
using partial pivoting, where
P
is a permutation matrix,
L
is
unit lower triangular, and
U
is upper triangular. Show your work!
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MATH 2070U Midterm
March 1st, 2018
Page 8 of 11
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MATH 2070U Midterm
March 1st, 2018
Page 9 of 11
5. Given
f
(
x
) =
x
3
+
4
x
2

10, the equation
f
(
x
) =
0 has a unique root in
[
1, 1.5
]
. With simple algebraic
manipulations we can show that both
x
=
g
1
(
x
)
:
=
1
2
(
10

x
3
)
1/2
and
x
=
g
2
(
x
)
:
= (
10
4
+
x
)
1/2
change
the given equation to the fixed point iteration. Knowing that
g
0
1
(
x
) =

3
4
x
2
(
10

x
3
)

1/2
and
g
0
2
(
x
) =

5
√
10
(
4
+
x
)
3/2
,
answer the following questions.
(a) [
1
/
2
mark] What is the value of
g
0
1
(
1.5
)
?
(a)
(b) [
1
/
2
mark] What is the value of

g
0
2
(
1.5
)

?
(b)
(c) [2 marks] Based on your result for parts (a) and (b) which of the above fixed point iterations
x
=
g
1
(
x
)
or
x
=
g
2
(
x
)
will converge more rapidly? Justify your answer!
(d) [4 marks] Based on your answer for part (c) choose the more efficient fixed point iteration and
perform 3 steps of the iteration starting at
x
0
=
1.5. Show your work!
(e) [2 marks] The actual value of the root is
x
*
=
1.3652 . Using the result of your computation in
part (d) compute the relative error at the third iteration. Show your work!
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MATH 2070U Midterm
March 1st, 2018
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