EEEN30002
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Three hours

Special instructions:
 Mathematical formulae tables supplied by the Examinations Office





THE UNIVERSITY OF MANCHESTER

Faculty of Science and Engineering
School of Electrical and Electronic Engineering





Numerical Analysis




22 January 2019
14:00 – 17:00


Answer all questions.
Electronic calculators may be used in accordance with the University regulations.





© The University of Manchester, 2019
EEEN30002
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Question 1
(a) The fraction 1/ 7 is expressed to four decimal places as 0.1429 . What is the
magnitude of the relative error? Express your answer as a simple fraction
where the numerator and denominator have no common factors.
[5 marks]
(b) Express 1/ 7 as a binary fraction, indicating clearly any recurring terms.
[5 marks]
(c) The floating point representation of the decimal number 1 is
hex3ff0 0000 0000 0000 .
(i) What is the floating point representation of the decimal number 2 ?
(ii) What is the floating point representation of the decimal number 0.5?
(iii) What is the floating point representation of the decimal number 2.5 ?
(iv) What decimal number does hex3ff0 0000 0000 0001 represent? You may
express your answer as a mixed fraction.
[2, 2, 2, 3 marks]
(d) A fixed-point algorithm is given as
1 2
5 4
:
6 3
k
k
k
x
x
x
   where k is an integer.
(i) If the algorithm converges, what value does it converge to?
(ii) Suppose the algorithm converges to *x . If 0 1x  , what is the magnitude of
the relative error of 1x with respect to *x ?
[3, 3 marks]
Total [25 marks]
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Question 2
A function ( )f x takes the following values:
x 0 2
1
1 5



2
1 5

1 2
2
1 5



2
1
1 5



2
( )f x 8 6 3 1 1 2 1

In addition, ( )f x is known to be continuous with only two zero crossings and only one
local minimum.

(a) Sketch a possible graph for ( )f x .
[6 marks]
(b) Suppose the two zero crossings occur at
1x and 2x ; i.e. 1( ) 0f x  and 2( ) 0f x 
with
1 2x x . What are the smallest intervals 1 1[ , ]a b and 2 2[ , ]a b where it is
known that 1 1 1a x b  and 2 2 2a x b  ?
[4 marks]
(c) What is the smallest interval [ , ]c d where it is known that *c x d  and ( *)f x is
the global minimum? What can you say about the numerical value of ( *)f x ?
[5 marks]
(d) Suppose you were to conduct a golden ratio search for the global minimum with
additional values of x and ( )f x . What would your next choice of x be?
[4 marks]
(e) Is the function ( )f x convex? Explain your answer.
[6 marks]
Total [25 marks]
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Question 3
(a) A 5 1 vector x is given by
Ax y
with
2 0 1 1 3
1 2 0 1 5
0 4 4 0 4
0 0 0 2 8
0 0 0 0 1
A
 
   
 
 
 
 
  
and
5
4
2
1
4
y
 
 
 
  
 
 
  
.
(i) Find 5x , the fifth element of x .
(ii) Find 4x , the fourth element of x .
(iii) Find all five elements of x using Gaussian elimination with partial pivoting
where appropriate, showing each step of the calculation.
[2, 2, 10 marks]
(b) Show that a Householder matrix, defined as
2
2
Tvv
H I
v
 
for some column vector v is both symmetric and orthonormal.
[6 marks]
(c) Find the QR factorization of the matrix
3 5
4 7
 
 
 

using Givens rotations.
[5 marks]
Total [25 marks]
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Question 4
A wave signal is given by
( ) sin14 2cos6y t t t   .
It is shown in Figure Q4.1 over the interval 0 1t  .

Figure Q4.1. Signal ( )y t over the interval 0 1t  .

(a) Show that the signal is periodic with period 1 s.
[5 marks]
(b) What is its rms value given as
1
2
rms
0
( )y y t dt  ?
It is not necessary to show the full calculation, but you should justify your
answer briefly.
[6 marks]






Question 4 continues over the page.

EEEN30002
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Question 4 continued
(c) An extension of Simpson's rule gives the integral of a function ( )f x over the
interval
0 nx x x  as
0
1 2
0
0
1,3,... 2,4,...
( ) ( ) ( ) 4 ( ) 2 ( )
3
n
n nx
n
n k k
x
k k
x x
f x dx f x f x f x f x
n
 
 
 
    
 
 
where ( )f x is evaluated at 1n equally spaced points 0 1, ,..., nx x x with n even.
(i) What approximation does this rule give for the rms when evaluated on the
interval 0 1t  when 6n  ?
(ii) Why might there be good reason not to trust the approximation for this
case?
[10, 4 marks]
Total [25 marks]














END OF EXAMINATION PAPER