Engg7302 Advanced Computational Techniques in Engineering Assignment 2: Numerical Linear Algebra

发布时间：2024-05-21

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Engg7302 Advanced Computational Techniques in Engineering

Assignment 2: Numerical Linear Algebra

Due date: 22 May 24 15:00

Where to submit: Submit the assignment report via the Blackboard Turnitin assignment submission system.

This assignment is worth 24% of the total marks for the course.

It aims to develop your programming skills in MATLAB, deepen your understanding of numerical linear algebra, and explore applications of numerical linear algebra. Your solutions to this assignment will be MATLAB program listings and relevant output, together with any other mathematical derivations, notes or explanations that aid understanding.

In marking the assignment, the following criteria will be applied:

- Correctness of the programs and mathematical calculations

- Clear and concise documentation, in the form of comments in the code, as to the approach being used and

- Relevance of the output in verifying the correctness of the program and in illustrating the solution.

- Some consideration will also be given to the efﬁciency of the solution.

PART A (This part is worth 16% of the total marks for the course)

Question 1

A signal, denoted as b, is collected at M sampling points, represented as b(1), b(2), …b(m), …,b(M). Your task is to strategically insert N components, denoted as x (x(1), x(2), …,x(n),…x(N)), into the system to alter the signal profile. The altered signal profile is represented as B=Ax+b.

The matrix A and vector b arestored in the files A.mat and b.mat respectively. In Matlab, you can access this data using the following commands: load A; load b.

You are asked to implement the following task:

Figure 1: Illustration of the relationship between signals x, b and B.

The signal B will satisfy the following condition:

| − 1|⬚ ≤∈, where, B0 is the mean value of B.

And ∈ (m) = 0.0001, m=1,2,…M. In addition, band B are both positive vectors. The range

of components x: |x(n) | ≤ 0.01 , n=1,2, …,N.

You are asked to minimise the 0-norm, 1-norm and 2-norm of vector x.

Prepare the report on the solution details and prepare the Matlab code.

Note:

To minimise the 0-norm and 2-norm of vector x, you are asked to use the function fmincon () in Matlab. As for the 0-norm minimisation, you need to find an approximation function as the objective function. For example, see:

https://link.springer.com/article/10.1007/s00034-022-02216-y

To minimise the 1-norm of vector x, you can use linprog(), you are asked to use the function linprog () in Matlab.

PART B (This part is worth 8% of the total marks for the course)

Question 2

Assessment Type: Application

Task Description:

The student needs to prepare a report (at least three pages) on real-world applications that use concepts learned in the linear algebra (LA) part.

The presentation should contain the following sections

(1) introduction;

(2) theory/methods;

(3) results and discussion/interpretation;

(4) conclusion and

(5) reference.