Module code and Title DTS104TC Numerical Methods

School Title School of Artificial Intelligence and Advanced Computing

Assignment Title Assignment 1

Submission Deadline June 2, 2023. 5pm (GMT+8)

Final Word Count -

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Scoring – For Tutor Use

Student ID

Stage of Marking Marker

Code Learning Outcomes Achieved （F/P/M/D）

(please modify as appropriate) Final

Score

` A B C `

1st Marker – red pen

Moderation

– green pen

IM

Initials The original mark has been accepted by the moderator (please circle as appropriate): Y / N

` Data entry and score calculation have been checked by another tutor (please circle): Y`

2nd Marker if needed – green pen

For Academic Office Use Possible Academic Infringement (please tick as appropriate)

Date

Received Days late Late Penalty ☐ Category A

Total Academic Infringement Penalty (A,B, C, D, E, Please modify where necessary) *_*

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☐ Category B
☐ Category C
☐ Category D
☐ Category E
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INSTRUCTIONS

1.The weighting of this assignment is 80% of the final mark.

2.The marking criteria sheet is provided as a supplementary document.

3.Your submission should only be in English.

4.Where required, Matlab code should be attached as .m files.

a)State the relevant .m file name in Answer Sheet to each question.

b)It is allowed to make use of multiple .m files as functions and inputs.

c)User-defined functions are allowed and all relevant functions of all questions should be submitted in a single folder. The final answers to each question should be displayed in an executable .m file named after each question.

d)You should add comments on functions, for loops, while loops and variables used in your .m files to get full marks.

e)In some questions, please show your process of derivation before you use your Matlab codes to get full marks.

5.Answers to questions should be typed on the Assignment1 Answer Sheet as Word files. The assignment must be submitted in a Zip file with your Answer Sheet and all .m files (Check all documents needed in the Zip file in your Assignment1 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 (20/100)

Consider the following equation: Equation (1)

(a)Assume the initial guesses of x are set as 1 and 2. Solving x for the following cases: y=0 for an absolute error of 0.01 using Bisection method using Matlab. Find out how many iterations are required to determine the value of x and fill out Table-1(a) in your Answer Sheet and the final answers should be computed and submitted in Matlab, using a file named AnswerOneA.m (3 marks)

(b)Set the initial guess as 1, solve x for the following case: y=10 and y=13 for an absolute error of 0.01 using Newton-Raphson method in Matlab. Fill out Table-1(b) in your Answer Sheet and the final answers should be computed and submitted in Matlab, using a file named AnswerOneB.m. Please check if the Nweton-Rapson method will be convergence before using it. (Check process should be written in your Answer Sheet) (7 marks)

(c)Assume the initial guesses of x are set as 3 and 4. While solving for x based on the initial guesses and using the Bisection method, find out how many iterations are required to determine the value of x to an absolute error of 0.01. Show and discuss your calculation step in your Answer Sheet. If you can not solve it by using Bisection method, try to solve this problem by using other methods. (10 marks)

Submission requirements:

All relevant Matlab code should be copied & pasted into the Answer Sheet.

Attach your Matlab code as .m files in your submission.

Question – 2 (20/100)

Consider the system of linear algebraic equations: Equation (2)

(a)Implement the Gauss-Seidel method in Matlab and compute the solutions for Equation (2) for a tolerance of 1e-3. Fill out Table-2 in your Answer Sheet and the final answers should be computed and submitted in Matlab, using a file named AnswerTwoA.m(14 marks)

(b)Discuss one major problem you may encounter when use Gauss-Seidel method and show how you may address it in your Answer Sheet. (Assume the coefficient matrix is diagonally dominant) (6 marks)

Submission requirements:

All relevant Matlab code should be copied & pasted into the Answer Sheet.

Include Matlab code as .m files as part of your submission.

Question – 3 (20/100)

Consider the the Matrix A below: Equation (3)

(a)Apply the power method to find the highest eigenvalue for the inverse of Matrix A in Matlab. Set the stopping criterion as 1% (Estimated Eigenvector need to be normalized). Fill out Table-3(a) in your Answer Sheet (Add rows if it is needed) and the final answers should be computed and submitted in Matlab, using a file named AnswerThree.m (15 marks)

(b)Take the reciprocal of the final eigenvalue estimates in question(a) and compare with the smallest eigenvalue calculated using eig() function for the Matrix A in Matlab. Fill out Table-3(b) in your Answer Sheet. (5 marks)

Submission requirements:

All relevant Matlab code should be copied & pasted in the section below.

Attach your Matlab code as .m files in your submission.

Question – 4 (20/100)

Consider the equation below: Equation(4)

(a)Implement 4th-Order Runge-Kutta Methods to calculate the integral of function g in the interval between 0 and 1. The initial condition of G(x=0)=0 is known. Compare the results obtained with a different step size and report your findings. Fill out Table-4 in your Answer Sheet and show your process of derivation, final answers should be computed and submitted in Matlab, using a file named AnswerFourA.m (10 marks)

(b)Implement an adaptive Runge-Kutta Method in Matlab. Present your result and state the merit of the method of choice in your Answer Sheet. The final answers should be computed and submitted in Matlab, using a file named AnswerFourB.m(10 marks)

Submission requirements:

All relevant Matlab codes should be copied and pasted on the Answer Sheet.

Attach Matlab code as .m files in submission

Question – 5 (20/100)

Evaluate the rational integral expression (5) below.

(a)Set a=0 and b=2, in Matlab, calculate the approximate solution of Expression (5) using Gaussian Quadrature method with n=5. Fill out Table-5(a) in your Answer Sheet and the final answers should be computed and submitted in Matlab, using a file named AnswerFiveA.m(4 marks)

(b)Consider that the actual integral in the situation of Q5(a) is 0.4762997573969109. Use Matlab to test how many segments are required by the Trapezoidal method to obtain a better accuracy than Gaussian Quadrature with n = 5? Fill out Table-5(b) in your Answer Sheet and the final answers should be computed and submitted in Matlab, using a file named AnswerFiveB.m (4 marks)

(c)Use Gaussian Quadrature method with n=5 to evaluate Expression (5) for a=0 and b=+∞ in Matlab. Write down your answer in your Answer Sheet. Fill out Table-5(c) in your Answer Sheet and the final answers should be computed and submitted in Matlab, using a file named AnswerFiveC.m (8 marks)

(d)Explain why Gaussian Quadrature method is superior to Trapezoidal method; In which situation can the Trapezoidal method obtain a precise results? Present answer in your Answer Sheet. (4 marks)

Submission Requirements:

All relevant Matlab codes should be copied and pasted on the Answer Sheet.

Attach Matlab code as .m files in submission

END OF QUESTIONS

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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.

C. Compute eigenvalues and eigenvectors by the power method.

D. Solve boundary value and initial problems to finite precision.

E. Develop quadrature methods for numerical integration.

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 Outstanding work that is at the upper limit of performance.

Work would be worthy of dissemination under appropriate conditions.

Mastery of advanced methods and techniques at a level beyond that explicitly taught.

Ability to synthesise and employ in an original way ideas from across the subject.

In group work, there is evidence of an outstanding individual contribution.

Excellent presentation.

Outstanding command of critical analysis and judgment.

B 70 - 80 First Excellent range and depth of attainment of intended learning outcomes.

Mastery of a wide range of methods and techniques.

Evidence of study and originality clearly beyond the bounds of what has been taught.

In group work, there is evidence of an excellent individual contribution.

Excellent presentation.

Able to display a command of critical thinking, analysis and judgment.

C 60 - 69 Upper Second Attained all the intended learning outcomes for a module or assessment.

Able to use well a range of methods and techniques to come to conclusions.

Evidence of study, comprehension, and synthesis beyond the bounds of what has been explicitly taught.

Very good presentation of material.

Able to employ critical analysis and judgement.

Where group work is involved there is evidence of a productive individual contribution

D 50- 59 Lower Second Some limitations in attainment of learning objectives but has managed to grasp most of them.

Able to use most of the methods and techniques taught.

Evidence of study and comprehension of what has been taught

Adequate presentation of material.

Some grasp of issues and concepts underlying the techniques and material taught.

Where group work is involved there is evidence of a positive individual contribution.

E 40 - 49 Third Limited attainment of intended learning outcomes.

Able to use a proportion of the basic methods and techniques taught.

Evidence of study and comprehension of what has been taught, but grasp insecure.

Poorly presented.

Some grasp of the issues and concepts underlying the techniques and material taught, but weak and incomplete.

F 0 - 39 Fail Attainment of only a minority of the learning outcomes.

Able to demonstrate a clear but limited use of some of the basic methods and techniques taught.

Weak and incomplete grasp of what has been taught.

Deficient understanding of the issues and concepts underlying the techniques and material taught.

Attainment of nearly all the intended learning outcomes deficient.

Lack of ability to use at all or the right methods and techniques taught.

Inadequately and incoherently presented.

Wholly deficient grasp of what has been taught.

Lack of understanding of the issues and concepts underlying the techniques and material taught.

Incoherence in presentation of information that hinders understanding.

G 0 Fail No significant assessable material, absent, or assessment missing a “must pass” component.

Marking Criteria For Questions and Sub Question with Matlab Code on the basis of 100%.