DTS104TC Numerical Methods Assignment 2

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DTS104TC Numerical Methods

Assignment 2

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, Newton-Raphson 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 Newton-Raphson 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 Newton-Raphson method set initial guesses as 2. Fill out Table-1(a), Table-1(b) and Table-1(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 Newton-Raphson method to solve this equation. Fill out Table-1(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 multiple-application 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 Table-2 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 multiple-application 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