2. [6 marks] Starting from the initial iterate
x
(0)
= 1
.
5
, use Newton’s method to find the next two iterates
x
(1)
and
x
(2)
approximating a solution of the equation
x
3
=
x
+ 2
.
2.
MATH 2070U
Midterm examination
Page 6 of 7
3. Perform the following numeric conversions.
•
Be sure to show your reasoning; no credit will be given for the answer alone.
•
Write your answers to at least three digits using the roundtonearest rule.
•
Use normalised scientific notation with the leading nonzero digit to the right of the radix point,
i.e., in the form
(0
.d
1
d
2
. . . d
t
)
β
×
β
e
where
e
is a suitable exponent,
d
k
∈ {
0
,
1
, . . . , β

1
}
are digits
base
β
and
d
1
6
= 0
is the leading nonzero digit.
(a) [2 marks] Convert
(13
.
25)
10
to base 2.
(a)
(b) [2 marks] Convert
(13
.
24)
8
to base 16.
(b)
(c) [2 marks] Convert
(
E
.
4
)
16
to base 10.
(c)
MATH 2070U
Midterm examination
Page 7 of 7
4. [5 marks] Find the LU factors of the matrix
A
=
4.
6

3
2
3

6
1
5
0
2
. Assume pivoting is not required.
MATH 2070U
Midterm examination
Page 3 of 7
1. [8 marks] Write in the space provided the output you would expect in an interactive M
ATLAB
session
as a result of entering the given statements.
•
For numerical values returned, assume
format short
, i.e., numerical values displayed rounded
to 5 decimal digits.
•
Where appropriate, be sure to distinguish row and column vectors.
•
If the expression fails to evaluate, write
ERROR
with a short comment explaining the error.
•
Assume the following statements have been executed, so the variables
A
,
B
,
C
, and
D
are available
in the M
ATLAB
workspace.
Exam solutions (Blue cover sheet)
]
]
]
]
]
]
]
]
4 ]
]
(b)
The operator * is interpreted as
matrix multiplication here. Since
B is a 2x2 matrix and C is a 3x2
matrix, the matrixmatrix product
is not defined.
??? Error using ==> mtimes
Inner matrix dimensions must agree.
MATH 2070U
Midterm examination
Page 4 of 7
A = [1,7; 4,0];
B = [4,1;7,4];
C = [6,2;3,1;5,8];
D = [8,1,4;1,7,4];
(c)
A./B.ˆ2
(c)
(d)
A
*
D  1
(d)
The matrixmatrix product A*D is welldefined
because A is a 2x2 matrix and D is a 2x3 matrix.
Subtracting 1 is expanded out elementwise, i.e.,
one is subtracted from every element of the
2x3 matrix A*D.
[ 14
51
25 ]
A*D1 = [
]
[ 31
5
15 ]
Here, we have to observe order
of operations: ^ has higher
precedence than / (division),
so we interpret this as A./(B.^2)
(rather than (A./B).^2 which has
a different numerical value).
[
1/16
7
]
A./B.^2 = [
]
[
4/49
0
]
[ 1
7 ]
A = [
]
[ 4
0 ]
[ 4
1 ]
B = [
]
[
7
4 ]
[
6
2
]
[
]
C = [
3
1
]
[
]
[
5
8
]
[
8
1
4 ]
D = [
]
[ 1
7
4 ]
MATH 2070U
Midterm examination
Page 5 of 7