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.

Question 1                        (This part is worth 15% of the total marks for the course)

There is a signal b collected at M sampling points(b(1), b(2), …b(m), …,b(M)), and you are asked to strategically place N components x (x(1), x(2), …,x(n),…x(N)) into the system to modify the signal profile. The modified signal profile is expressed as  B=Ax+b.

The matrix A and vector b are stored in files: A.mat and b.mat, and in Matlab, you can access the data as follows: load A; load b.

You are asked to implement the following tasks:

Q1(a):

Use the Singular Value Decomposition (SVD) technique to find x (consider low-rank approximation r=200).

Based on your SVD results, find the condition number of A.

(Note- SVD can be calculated by Matlab built-in function).

Q1(b):

Use the regularisation approach (regularised linear least squares) to find x.

Select different regularisation parameters 入 =

1e−10, 1e−9, 1e−8, 1e−7, 1e−6, 1e−5, 1e−4, 1e−3, 1e−2

Plot an L-curve figure that shows the relationship between 2-norm(x) and 2-norm(Ax- b) with different regularisation parameters .

(Note – refer to lecture notes (LA9, Least square problems) .

Q1(c):

The signal B will satisfy the following condition: |B − B0 | ≤ B0 ∈ , where, B0 is the mean value of B. And ∈ (m) = 1e−4, m=1,2,…M. In addition, b and B are both    positive vectors. The range of components x: 0≤x(n) ≤ 5e−3, n=1,2,…,N.

Write a Matlab code to minimise the 1-norm of vector x.

Note:  please consider the function linprog() in Matlab;

Question 2                            (This part is worth 9% of the total marks for the course)

Assessment Type: Application

Task Description: