DTS104TC Numerical Methods
School of Artificial Intelligence and Advanced Computing
Assignment 2
May 31, 2024. 5pm (GMT+8)
LEARNING OUTCOMES
This assessment tests your ability to:
A. Apply numerical methods in a number of different contexts.
B. Solve systems of linear and nonlinear algebraic equations to specified precision.
E. Develop quadrature methods for numerical integration.
INSTRUCTIONS
1. The weighting of this assignment is 20% of the final mark.
2. The marking criteria sheet is provided as a supplementary document.
3. Your submission should only be in English.
4. When you fill out Tables in your Answer Sheet, you can add or delete rows if it is needed.
5. Answers to all questions in this assignment should be written on papers. You need to take photos of your answers and paste these photos to Assignment2 Answer Sheet then save the Answer Sheet as Word files. The assignment must be submitted in a Zip file with your Answer Sheet (Check all documents needed in the Zip file in your Assignment2 Answer Sheet) via Learning Mall Online to the correct drop box. Only electronic submissions are accepted and no hard copy submissions are permitted.
6. All students must download their file and check that it is viewable after submission. Documents may become corrupted during the uploading process (e.g. due to slow internet connections). However, students themselves are responsible for submitting a functional and correct file for their assessments.
Question – 1 (45/100)
Consider the following equation: Equation (1), assuming stability is not an issue and all methods will converge.
(a) Compare and contrast Bisection method, NewtonRaphson method and Secant method based on the number of initial guesses needed, rate of convergence and whether the required evaluation of the differentiation can be obtained. (9 marks)
(b) Use Bisection method, Secant method and NewtonRaphson method to calculate a root of equation (1), you should calculate by hands with the stopping criterion set to 5%. For Bisection and Secant Method, set initial guesses as 0.4 and 1. For NewtonRaphson method set initial guesses as 2. Fill out Table1(a), Table1(b) and Table1(c) and type your calculations for each iterations in your Answer Sheet. (24 marks)
(c) It is known that the equation has a unique root in the interval [2,3]. Please choose a suitable initial guess and use the NewtonRaphson method to solve this equation. Fill out Table1(d) and type your calculations for each iterations and the reason for choosing this initial guess in your Answer Sheet. (12 marks)
Question – 2 (28/100)
Evaluate the the rational integral expression (2).
(a) Evaluate Expression(2) analytically by hands. Type your calculations in your Answer Sheet; (5 Marks)
(b) Use multipleapplication of the trapezoidal rule (n=4) to evaluate Expression (2) by hands. Type your calculations in your Answer Sheet; (5 marks)
(c) Use single application of the Simpson’s 1/3 rule to evaluate Expression (2) by hands. Type your calculations in your Answer Sheet; (5 marks)
(d) Use single application of the Simpson’s 3/8 rule to solve evaluate Expression (2) by hands. Type your calculations in your Answer Sheet. (5 marks)
(e) Calculate the relative error based on analytical solution. Fill out Table2 and type your calculations in your Answer Sheet. (8 marks)
Question – 3 (27/100)
Evaluate the the rational integral expression (3).
(a) Use single application of the trapezoidal rule to solve Equation (3) by hands (round off to 2 decimal places). Type your calculations in your Answer Sheet; (4 marks)
(b) Use multipleapplication of the trapezoidal rule (n=2) to evaluate Expression (3) by hands (round off to 2 decimal places). Type your calculations in your Answer Sheet; (4 marks)
(c) Based on the answers of question (a) and (b), use the Romberg integration method to calculate the value for the integral by hands. Type your calculations in your Answer Sheet; (4 marks)
(d) Try to prove that the answer in question (c) is closer to the true value of the integral than the answer in question (a) (without calculating the true value of the integral); (15 marks)
MARKING CRITERIA
The following table indicates what is expected for each classification category, highlighting generic marking criteria that bring together expectations in performance for each percentage (or alphabetical) band and the criteria that need to be satisfied.
Generic Marking Criteria
Grade

Point Scale


Criteria to be satisfied

A

81+

First

Ø Outstanding work that is at the upper limit of performance.
Ø Work would be worthy of dissemination under appropriate conditions.
Ø Mastery of advanced methods and techniques at a level beyond that explicitly taught.
Ø Ability to synthesise and employ in an original way ideas from across the subject.
Ø In group work, there is evidence of an outstanding individual contribution.
Ø Excellent presentation.
Ø Outstanding command of critical analysis and judgment.

B

70  80

First

Ø Excellent range and depth of attainment of intended learning outcomes.
Ø Mastery of a wide range of methods and techniques.
Ø Evidence of study and originality clearly beyond the bounds of what has been taught.
Ø In group work, there is evidence of an excellent individual contribution.
Ø Excellent presentation.
Ø Able to display a command of critical thinking, analysis and judgment.

C

60  69

Upper Second

Ø Attained all the intended learning outcomes for a module or assessment.
Ø Able to use well a range of methods and techniques to come to conclusions.
Ø Evidence of study, comprehension, and synthesis beyond the bounds of what has been explicitly taught.
Ø Very good presentation of material.
Ø Able to employ critical analysis and judgement.
Ø Where group work is involved there is evidence of a productive individual contribution

D

50 59

Lower Second

Ø Some limitations in attainment of learning objectives but has managed to grasp most of them.
Ø Able to use most of the methods and techniques taught.
Ø Evidence of study and comprehension of what has been taught
Ø Adequate presentation of material.
Ø Some grasp of issues and concepts underlying the techniques and material taught.
Ø Where group work is involved there is evidence of a positive individual contribution.

E

40  49

Third

Ø Limited attainment of intended learning outcomes.
Ø Able to use a proportion of the basic methods and techniques taught.
Ø Evidence of study and comprehension of what has been taught, but grasp insecure.
Ø Poorly presented.
Ø Some grasp of the issues and concepts underlying the techniques and material taught, but weak and incomplete.

F

0  39

Fail

Ø Attainment of only a minority of the learning outcomes.
Ø Able to demonstrate a clear but limited use of some of the basic methods and techniques taught.
Ø Weak and incomplete grasp of what has been taught.
Ø Deficient understanding of the issues and concepts underlying the techniques and material taught.
Ø Attainment of nearly all the intended learning outcomes deficient.
Ø Lack of ability to use at all or the right methods and techniques taught.
Ø Inadequately and incoherently presented.
Ø Wholly deficient grasp of what has been taught.
Ø Lack of understanding of the issues and concepts underlying the techniques and material taught.
Ø Incoherence in presentation of information that hinders understanding.

G

0

Fail

Ø No significant assessable material, absent, or assessment missing a “must pass” component.
