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程序代写案例-EEEN30002

时间：2021-01-07

EEEN30002

1 of 6

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

2 of 6

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]

EEEN30002

3 of 6

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]

EEEN30002

4 of 6

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]

EEEN30002

5 of 6

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

6 of 6

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

1 of 6

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

2 of 6

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]

EEEN30002

3 of 6

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]

EEEN30002

4 of 6

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]

EEEN30002

5 of 6

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

6 of 6

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