0304 Exam

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Liverpool John Moores UniversitySCHOOL OF COMPUTING AND MATHEMATICAL SCIENCES

Semester 2 Examinations 2004

CMPMA2022

Numerical Methods and Linear Algebra

Duration 2 Hours

Instructions to candidates

Do not open this question paper until you have been told to do so by the invigilator.

The figure in [] denotes the number of marks available for that question or part of question.

There are 8 questions. Answer 5 questions.

Questions carry 20 marks each. The total number of marks available is 100.

You have the use of the Software DERIVE 5 and Matlab. You may also use any Derive

functions that you have developed throughout the course. You may not use a printer to print

out any mathematical expressions or graphs.

Where a question states that Derive and Matlab are not to be used, marks will be lost if

working is not shown.

Any results or graphs that DERIVE 5 produces, on screen, that are relevant to your

examination work should be written or sketched in your answer book.

In answering some of the questions in this examination Derive may generate expressions that

are too large to copy down into your answer booklet. In these cases, save the expressions in adfw file (e.g. q4iii.dfw) onto the floppy disk provided. Ensure that the expression numbers

are clearly referenced in your answer booklet and that you hand the floppy disk to the

invigilator at the end of the examination with your script. Also send me an email

([email protected]), at the end of the exam, with the dfw files as attachments as a

backup precaution.

You may also use any calculator and its memory need not be erased before the examination.

The sending or the reading of email and use of the internet/intranet during this examination is

prohibited.

CMPMA2022/2004 Page 1/6 Set: Dr T.A.Etchells
mailto:[email protected]:[email protected]

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1. (i) Describe, with the aid of a sketch, the Newton-Raphson iterative method for

solving single variable equations and develop the Newton-Raphson iterative

formula. Describe a situation in which the Newton-Raphson method will fail

to produce a solution for a particular equation. [5]

(ii) Show that the equation 2 sin 4xx e x = has at least one solution in theinterval [0,1]. (A graph alone will not gain full marks)

[4]

(iii) Establish the Newton-Raphson iterative formula for the

equation 2 sin 4xx e x = .

[3]

(iv) Using a starting value ofx0 1= , find each of the Newton-Raphson iterates

1 2 3 4 5, , , andx x x x , to 10 significant figures.

[2]

(v) Write down the solution to the equation 2 sin 4xx e x = , correct to 6

significant figures. Confirm that this solution is correct to 6 significantfigures.

[3]

(vi) Explain why x is not a good choice of starting value the Newton

Raphson method for the equation

00=

2 sin 4xx e x = .

[3]

Total [20]

2. (i) Describe the power method for finding the dominant eigenvector and eigenvalueof a square matrixA. [5]

(ii) Using a starting vector of [1,1,1]=0v , apply the power method once to the

matrix

1 3 1

2 3 42 4 5

to obtain and approximation to the dominant eigenvector . [3]1

v

(iii) Continue to apply the power method until the approximations to thedominant eigenvector converge to 3 decimal places. Write down the value of

the dominant eigenvalue correct to 3 decimal places. [8]

(iv) Use the dominant eigenvector found in (iii) to estimate the dominant

eigenvalue. [4]Total [20]


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3. (a) (i) Describe fully, with reference to a sketch, the modified Euler method for

solving the first order differential equation ( , )dy

f x ydx

= , with initial

conditions . [5]0

( )y x y= 0

(ii) Given the differential equation 2xdy

e xydx

=

0.1x

with initial condition .

Showing all the steps, use the modified Euler method to find an

approximation to the value ofy when

(0) 1y =

= , using a step size h = 0.1.[5]

(b) Given the system of first order differential equations:

2

2dx

y tdt

dyxy t

dy

= +

= +

subject to the initial conditions 0 0 01, 1, 0x y t= = = .

Use the modified Euler method, with a step size of 0.1, to find an

approximate solution to this system of differential equations when t .0.1=[10]

Total [20]

4. (a) (i) Given that Vis a vector space, write down the two conditions that prove

that Wis a subspace ofV. [2]

(ii) Given that all 2x2 matrices that have real elements form a vector spaceM,

show that matrices of the form

0

0

a

b

where a b, \ , form a subspace ofM. [3]

(b) Determine whether ,2= + +1v i j 2 = +v i and 3 2 3= + +v i j span the

vector space\ . [6]3

(c) (i) Write down the condition that the set of vectors { , is a linearly

dependent set. [2]

1 2 3, }v v v

(ii) Show that the set of vectors { }(0,3,1, 1),(6,0,5,1),(4, 7,1,3) 4

form a

linearly dependent set in . [7]\

Total [20]


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5.

(i) Using the function M(a,b,n) in the file Midpoint.mth that can be found in

the folderL:\MA200, generate to 10 significant digits 5 approximations of

the integral

1

20

sin

1

xdx

x+,

using the composite mid-point rule with 2, 4, 8 and 16 strips respectively.

[6]

(ii) Use the knowledge that the midpoint rule has order of convergence to

apply Rombergs method once to the 5 approximations calculated above. [5]

2h

(iii) State the order of convergence of these new approximations. [2]

(iv) Apply Rombergs method a further 2 times to find a better approximation to

1

.20 1xe

dxx+

[7]

Total[20]

6. (i) Describe theshooting methodfor the solution of boundary value problems

for second order ordinary differential equations. [5]

(ii) Recast the second order differential equation given below into two first

order differential equations.

22

24 5 sinx

d y dyx y e

dx dx

+ = x

[4]For the next part of the question use the Derive 5 Runge-Kutta order four

function RK().

(iii) Given that y and(0) 0= (1) 0y = , use the shooting methodwith a step size of0.05 to find a numerical solution to this differential equation in the

region [0,1]x such that (1) 0y = , where 0.01 < .

Save this solution in a dfw file as q3iii.dfw on the floppy disk provided, do

not attempt to copy it out.

Provide a sketch of the solution. [11]

Total [20]


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7.

Given the matrix

2 6 2

3 8 0

4 9 2

A

=

(i) Write down the elementary matrix that will perform the row

operation

1E

11

2

R . [2]

(ii) Write down the inverse of the elementary matrix . [1]1

E

(iii) Find the other elementary matrices and , such that the product

will reduceA to row-echelon form and determine the row-

echelon form ofA. [8]

2 3, ,E E E

4

5E

5 4 3 2 1E E E E E A

(iv) Find theLUdecomposition ofA, whereL is a lower triangular matrix and U

is an upper triangular matrix (in row-echelon form). [4]

(v) Use theLUdecomposition ofA to solve the linear system:

1

2

3

2

2

3

x

A x

x

=

[5]

Total [20]

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8. The use of Derive or MATLAB is notallowed in the question, working

must be shown to gain full marks.

(i) Write down the value of the determinant

1 2 3

0 3 2

0 0 2

[2]

(ii) Reduce the determinant

0 1 5

3 6

2 6 1

9

to row-echelon form. [8]

(iii) Use the row-echelon form found in (b) to evaluate

0 1 5

3 6

2 6 1

9

[3]

(iv) Using the answer in (c), evaluate

0 2 10

3 6 9

6 18 3

[4]

(v) Write down the value of

0 2 5 0 3 6

3 6 9 2 6 18

6 18 3 5 9 13

[3]

Total [20]

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0304 Exam